Before going to learn about Solving Quadratic Equations, first recall a few facts about the quadratic equations. The word quadratic originated from the word quad and its meaning is “square”. It means that the quadratic equation has a variable raised to 2 as the greatest power term. The standard form of a quadratic equation is given by the equation ax2 + bx + c = 0, where a ≠ 0.
We saw that quadratic equations can represent many real-life situations. Now that we know what quadratic equations are, let us learn about the definition of solving quadratic equations, and the different methods to solve them. Here we will try to develop the Quadratic Equation Formula and other methods of solving the quadratic equations. But the solving quadratic equations by factoring is the most popular method. You learn all the methods in detail here along with all 10th Grade Math Concepts all here.
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What is meant by Solving of Quadratic Equations | Solving of Quadratic Equations-Definition
It is defined as, that any value(s) of x that satisfies the equation is known as a solution (or) root of the equation, and the process of finding the values of x which satisfy the equation ax2 + bx + c = 0 is known as solving quadratic equations.
Methods of Solving a Quadratic Equation | How to Solve Quadratic Equations?
Solving quadratic equations means finding a value (or) values of the variable which satisfies the equation. The value that satisfies the quadratic equation is called its roots (or) solutions (or) zeros. Hence the degree of the quadratic equation is 2, it can have a maximum of 2 roots. But how do find them if they are not given? The different methods of solving quadratic equations are:
1. Solving quadratic equations by factoring
2. Solving quadratic equations by using completing the square
3. Solving quadratic equations by graphing
4. Solving quadratic equations by quadratic formula
Solving Quadratic Equations by Factoring
This method is one of the most famous and simplest methods used to solve a quadratic equation and certain quadratic equations can be factorized. If we have done correctly will give get two linear equations in x. Hence, from that equations, we will get the value of x. The step-by-step process of solving quadratic equations by factoring is explained below along with an example we will solve the equation is x2-3x + 2 = 0.
- Step 1: First, we get the equation into a standard form.
- Step 2: Then factorize the quadratic equation.
- Step 3: By zero product property, set each of the factors to zero.
- Step 4: Now, solve each of the above equations.
Example: Solve the equation step by step using the factoring method. The equation is x2-3x + 2 = 0.
Solution: Given that,
- Step1: Get the equation into standard form. i.e., Get all the terms to one side (usually to the left side) of the equation such that the other side will be 0. The equation x2 – 3x + 2 = 0 is already in standard form.
- Step 2: Factor the quadratic expression. Then we will get as (x – 1) (x – 2) = 0.
- Step 3: In zero product property, set each of the factors will be zero that is x – 1 = 0 (or) x – 2 = 0
- Step 4: Solve each of the above equations. x = 1 (or) x = 2
Thus, the solutions of the quadratic equation x2 – 3x + 2 = 0 are 1 and 2. This method is only applicable when the quadratic expression is factorable. If it is not factorable, then we can use one of the other methods. Similar to the quadratic equations we have a solution for linear equations, which are used to solve linear programming problems.
Solving Quadratic Equations by using Completing the Square
In this method, Completing the square means to write the quadratic expression as ax2+bx+c into the form a(x – h)2 + k (it is also known as vertex form), where h = -b/2a and ‘k’ can be obtained by substituting x = h in ax2 + bx + c. The step-by-step process of solving the quadratic equations by completing the square is given below, along with an example where we are going to find the solutions of the equation 2x2 + 8x = -3.
- Step 1: First, we get the equation into a standard form.
- Step 2: Now, complete the square on the left side.
- Step 3: Solve it now, we get the value of x.
Example:
Solve the equation using the complete square method. The equation is 2x2+8x = -3.
Solution:
Given that the equation is 2x2+8x = -3
Step 1: Initially, we get the equation into standard form.
Now adding 3 on both sides, we get 2x2 + 8x + 3 = 0.
Step 2: Complete the square on the left side, then we get 2(x + 2)2-5 = 0.
Step 3: Now, Solve it for x. (We will take the square root on both sides along the way).
Next, Adding 5 on both sides, that is 2 (x + 2)2 = 5.
Dividing both sides by 2,
(x + 2)2 = 5/2
Taking square root on both sides,
x + 2 = √(5/2) = √5/√2 · √2/√2 = √10/2
Let Subtracte 2 from both sides,
x = -2 ± (√10/2) = (-4 ± √10) / 2
Thus, the roots of the quadratic equation 2x2+8x = -3 are (-4 + √10)/2 and (-4 – √10)/2.
Solving Quadratic Equations by Graphing
To solve the quadratic equations by using graphing, first, we have to graph the quadratic expression (when the equation is in the standard form) either manually or by using a graphing calculator. Then the x-intercepts of the graph (the point(s) where the graph cuts the x-axis) are nothing but the roots of the quadratic equation. The process of solving quadratic equations by graphing is explained in steps along with an example, and we are going to solve the equation 3x2 + 5 = 11x.
- Step 1: Initially, we get the equation into the standard form.
- Step 2: Graph the quadratic expression which is on the right side.
- Step 3: Identify the X-intercepts.
- Step 4: Next, the x-coordinates of the x-intercepts are nothing but the roots of the quadratic equation.
Example:
Solve the equation is 3x2 + 5 = 11x.
Solution:
Given that,
Step 1: Initially, we get the equation into the standard form.
First, subtracting 11x from both sides, 3x2 – 11x + 5 = 0.
Step 2: Graph the quadratic expression which is on the left side.
Graph the quadratic function y = 3x2 – 11x + 5 either manually or using a graphing calculator (GDC).
Step 3: Identify the x-intercepts.
For solving quadratic equations by graphing, the quadratic expression has to be graphed and identify the x-intercepts.
Step 4: Now, the x-coordinates of the x-intercepts are nothing but the roots of the quadratic equation.
Thus, the solutions of the quadratic equation 3x2 + 5 = 11x are 0.532 and 3.135. By observing the above example, we can see that the graphing method of solving quadratic equations may not give the exact solutions (i.e., it gives only the decimal approximations of the roots if they are irrational}. i.e., if we solve the same equation using completing the square, we get x = (11 + √61) / 6 and x = (11 – √61) / 6.
But we will not get the exact roots by the graphing method. If the graph does not intersect the x-axis at all, it means that the quadratic equation has two complex roots that is the graphing method is not useful to find the roots if they are complex numbers. We can use the quadratic formula to find any type of the root’s value (it will be explained in the next section)
Solving Quadratic Equations by Quadratic Formula
As we have already seen, the previous methods for solving the quadratic equations have some limitations such as the factoring method is useful only when the quadratic expression is factorable, the graphing method is useful only when the quadratic equation has real roots, etc.
But solving quadratic equations by quadratic formula overcomes all these limitations and is useful to solve any type of quadratic equation. Here is the step-by-step explanation of solving quadratic equations by quadratic formula along with an example where we will be finding the solutions of the quadratic 2x2 = 3x – 5.
- Step 1: First, we get the equation into a standard form.
- Step 2: Now, compare the equation with ax2+bx+c=0 and then find the values of a,b and c.
- Step 3: Substitute the values into the quadratic formula which says x = [-b ± √(b² – 4ac)] / (2a).
- Step 4: Now, simplify it, we get the x value.
Example:
Solve the equation using the quadratic formula. The equation is 2x2 – 3x + 5 = 0.
Solution:
As given in the question. the equation is 2x2 – 3x + 5 = 0.
- Step 1: Get into the standard form. Then the above equation becomes 2x2 – 3x + 5 = 0.
- Step 2: Compare the equation with ax2 + bx + c = 0 and find the values of a, b, and c. Then we get the value of a is 2, b is -3. and c is 5.
- Step 3: Substitute the values into the quadratic formula which says x = [-b ± √(b² – 4ac)] / (2a). Then we get
x = [-(-3) ± √((-3)² – 4(2)(5))] / (2(2) - Step – 4: Simplify it, then the value of x is
x = [ 3 ± √(9 – 40) ] / 4 = [ 3 ± √(-31) ] / 4 = [ 3 ± i√(31) ] / 4
Thus, the roots of the quadratic equation 2x2 = 3x – 5 are [ 3 + i√(31) ] / 4 and [ 3 – i√(31) ] / 4. In a quardratic formula, the expression of b² – 4ac is called as discriminant (which is denoted by D). i.e., D = b² – 4ac. This will used to determine the nature of roots of the quadratic equation.
Nature of Roots Using Discriminant
- If D > 0, then the equation ax2 + bx + c = 0 has two real and distinct roots.
- If D = 0, then the equation ax2 + bx + c = 0 has only one real root.
- If D < 0, then the equation ax2 + bx + c = 0 has two distinct complex roots.
Thus, using the discriminant, we can find the number of solutions to quadratic equations without actually solving them. Apart from these methods, there are a few other methods that are used only in specific cases i.e., when the quadratic equation has missing terms like that, the below explained:
Solving Quadratic Equations Missing b
In a quadratic equation ax^2 + bx + c = 0, if the term with b is missing then the equation becomes ax^2 + c = 0. Now, we can solve this by taking square root on both sides. The below explained the process with examples.
The equation is x^2 – 4 = 0
⇒ x^2 = 4 ⇒ x = ±√4 ⇒ x = ± 2
So, the equation roots are 2 and -2.
The another example is, x^2 + 36 = 0
⇒ x^2 = -36 ⇒ x = ±√(-36) ⇒ x = ± 6i
Thus, the roots of the equation are 6i and -6i (note that these are imaginary numbers (or) complex numbers).
Solving Quadratic Equations Missing c
In a quadratic equation ax^2 + bx + c = 0, if the term with c is missing then the equation becomes ax^2 + bx = 0. To solve this type of equation, we simply factor x out from the left side, set each of the factors to zero, and solve them. The process will explained in below with examples:
The equation is x^2 – 5x = 0
⇒ x (x – 5) = 0 ⇒ x = 0; x – 5 = 0 ⇒ x = 0; x = 5
So, the equation roots are 0 and 5.
Next, the equation is x^2 + 11x = 0 ⇒ x (x + 11) = 0
⇒ x = 0; x + 11 = 0 ⇒ x = 0; x = -11
Hence, the roots of the equations are 0 and -11.
Solving Quadratic Equations Examples with Answers
Problem 1: The length of a park is 5 ft less than twice its width. If its area is 250 square feet, find the dimensions of the park?
Solution:
The data is as given in the question,
Assume that, the width of the park is x ft.
Then the length of the park is (2x – 5) ft.
The area of a park is 250 sq. ft
So, the length × width is 250.
Substitute the values, it will be (2x – 5)x = 250.
2x^2 – 5x – 250 = 0.
So, this is a word problem that is related to solving quadratic equations. Now, let us solve this quadratic equation by using the factoring method.
In this, the value of a is 2, b is -5, and c is -250.
So, a x c is 2(-250) = -500.
Next, the two numbers whose sum is -5 and whose product is -500 are -25 and 20. So we can split the middle term using these two numbers.
2x^2 – 25x + 20x – 250 = 0
x(2x – 25) + 10 (2x – 25) = 0
(2x – 25) (x + 10) = 0
2x – 25 = 0 i.e., x + 10 = 0
The value of x is x = 25/2 = 12.5 (or) x = -10.
x = 12.5, x value cannot be in negative.
So the width is 12.5 ft and the length is (2x – 5) ft = 2(12.5)-5 = 20 ft.
Thus, the dimensions of the park are 20 ft × 12.5.
Problem 2: Let the two positive consecutive numbers is 156. Find the value of two numbers?
Solution:
Given that,
The two positive consecutive numbers product is 156.
Now, we will find the values of two numbers.
Assume that the two consecutive numbers be x and x + 1. Then the equation is,
x (x + 1) = 156
x^2 + x – 156 = 0
Now, solve this quadratic equation by the factoring method.
In this a = 1, b = 1 and c = -156.
So, the ac is 1(-156) = -156.
The two numbers whose sum is 1 and whose product is -156 are 13 and -12. So we can split the middle term using these two numbers.
i.e., x^2 + 13x – 12x – 156 = 0
x (x + 13) – 12 (x + 13) = 0
(x + 13) (x – 12) = 0
The value of x + 13 = 0, and x – 12 = 0
Therefore, x = -13 (or) x = 12
we know, x is positive, x cannot be in negative i.e.,-13. So the value of x is 12.
Thus, the required consecutive numbers are 12 and 13 (12 + 1).
FAQs on Solving Quadratic Equations
1. What is a quadratic equation?
We Simply say that a quadratic equation is an equation of degree 2, which means that the highest exponent of this function is 2. Moreover, the standard quadratic equation is ax^2 + bx + c, where a, b, and c are just numbered and ‘a’ cannot be 0. An example of quadratic equation is 3x^2 + 2x + 1.
2. What are the different methods by which you can solve quadratic equations?
There are various methods by which you can solve a quadratic equation such as factorization, completing the square, quadratic formula, and graphing. All these are the four general methods that we can use to solve a quadratic equation.
3. What are the three forms of a quadratic function?
The three functions are listed below which can be written as:
1. Standard Form: y = ax2 + bx + c, where a, b, and c are just numbers.
2. Factored Form: y = (ax + c) (bx + d) where a, b, and c are just numbers.
3. Vertex Form: y = a (x + b)2 + c, and here also a, b, and c are numbers.
4. What is the formula for solving quadratic equations?
The general quadratic equation formula is “ax2 + bx + c”. In this formula, a, b, and c numbers, are the numerical coefficient of the quadratic equation, and ‘a’ is not zero a 0.
5. How is the Factored Form Helpful in Solving Quadratic Equations?
If the quadratic expression that is in the standard form of quadratic expression in it is factorable, then we can just set each factor to zero, and solve them. Thus, the solutions are nothing but the roots of a quadratic equation.