The equation of second degree is known as a quadratic equation. Quadratic is nothing but square because the highest degree is square. In order to solve the problems first, we have to write the equation in the standard form. The students of 9th grade can know clearly about the methods of solving the quadratic equations with the help of this page. You can score the best marks in the exams. Let us discuss in detail the ways of solving quadratic equations with questions and answers.
3 Methods of Solving Quadratic Equations
There are three different methods of solving a quadratic equation. They are
i. Factorization method
ii. By completing the square
iii. By using the quadratic equation formula
Solving Quadratic Equations by Factoring Method
There are some steps for solving the quadratic equation factorization method.
Step 1: First write the given equation in the standard form.
Step 2: Now factorize the quadratic equation on the left side.
Step 3: Left side of the equation will be the product of two linear factors.
Step 4: Equate each of the linear factors to zero and solve for the values of x.
Step 5: The value of x will be the roots of the equation.
Example:
Solve the equation 3x² + 5x – 2 = 0
Solution:
Write the given equation in the standard form ax² + bx + c = 0
3x² + 5x – 2 = 0
3x(x + 2) -1(x + 2) = 0
(3x – 1)(x + 2) = 0
3x – 1 = 0 or x + 2 = 0
3x = 1
x = 1/3
x + 2 = 0
x = -2
Thus x = 1/3, -2 are the roots of the equation
Solving Quadratic Equations by Completing the Square Method?
The steps of solving the quadratic equation by completing the square are as follows,
Step 1: First we have to write the standard form of the quadratic equation.
Step 2: Divide both sides of the equation by the coefficient of x².
Step 3: Shift the constant term to the Right-hand side of the equation.
Step 4: Add the square of half of the coefficient of x to both sides.
Step 5: Write the L.H.S as a complete square and simplify the R.H.S.
Step 6: At last take the square root on both sides and solve for the value x.
Example:
Solve the equation 3×2 + 4x – 15 = 0 by completing the square?
Solution:
Write the given equation in the standard form ax² + bx + c = 0
x² + 4/3x – 5 = 0
Adding the square of one half of the coefficient of x
x² + 4/3x + (4/6)² = 5 + (4/6)²
(x + 4/6)² = 5 + (16/36) = 196/36
Taking square root of both sides and solve x.
x + 4/6 = 14/6
x + 4/6 = ±7/3
x + 4/6 = 7/3
x = 7/3 – 4/6
x = 10/6
x = 5/3
x + 4/6 = -7/3
x = -7/3 – 4/6
x = -18/6
x = -3
Hence the solution is {-3, 5/3}
Solving Quadratic Equations by Using the Quadratic Formula
In order to solve the quadratic equation, we have to follow some set of rules.
Step 1: First of all we have to write the equation in the standard form.
Step 2: Identify a, b, c values by comparing the equation with the standard form.
Step 3: Put the values a, b, c in the quadratic formula.
(α, β) x = [-b ± √(b² – 4ac)]/2a
Example:
Solve the equation x² + 7x + 12 = 0?
Solution:
Write the given equation in the standard form ax² + bx + c = 0
x² + 7x + 12 = 0
a = 1
b = 7
c = 12
x = [-b ± √(b² – 4ac)]/2a
x = [-7 ± √(7² – 4.1.12)]/2.1
x = [-7 ± √(49 – 48)]/2
x = [-7 ± √1]/2
x = [-7 + 1]/2 or x = [-7 – 1]/2
x = -6/2
x = -3
x = -8/2
x = -4
Thus the value of x is -3, -4
Do Check: Introduction to Quadratic Equation
Different Methods of Solving Quadratic Equations Examples
Example 1.
Solve the equation 2x² + 3x = 2 by factorization method?
Solution:
2x² + 3x = 2
Factorize the left hand size
2x² + 3x – 2 = 0
(2x – 1) (x – 2)
Equate each of the linear factors to zero.
2x – 1 = 0 or x + 2 = 0
2x = 1 or x = -2
x = ½
x = ½ , -2 are the roots of the equation
Solution set = {½ , -2}
Example 2.
Solve the equation x² + 8x + 16 = 0 by using quadratic formula?
Solution:
Write the given equation in the standard form ax² + bx + c = 0
x² + 8x + 16 = 0
a = 1
b = 8
c = 16
x = [-b ± √(b² – 4ac)]/2a
x = [-8 ± √(8² – 4.1.16)]/2.1
x = [-8 ± √(64 – 64)]/2
x = [-8]/2
x = -4
Thus the value of x is -4
Example 3.
Solve the equation 2x² + 9x + 7 = 0?
Solution:
Write the given equation in the standard form ax² + bx + c = 0
2x² + 9x + 7 = 0
a = 2
b = 9
c = 7
x = [-b ± √(b² – 4ac)]/2a
x = [-9 ± √(9² – 4.2.9)]/2.2
x = [-9 ± √(81 – 72)]/4
x = [-9 ± √9]/4
x = [-9 + 3]/4 or x = [-9 – 3]/4
x = -6/4
x = -3/2
x = -12/4
x = -3
Thus the value of x is -3/2, -3
FAQs on Methods of Solving Quadratic Equations
1. Which method is best for solving quadratic equations?
Factoring or quadratic formula is the best method for solving quadratic equations.
2. What are the 3 methods of solving quadratic equations?
There are three basic methods for solving quadratic equations are factoring, using the quadratic formula, and completing the square.
3. How to factor a quadratic equation?
i. Expand the expression and simplify all fractions if needed.
ii. Move the terms to the left-hand side of the equal sign.
iii. Now, Factorize the equation by breaking down the middle term.
iv. Equate each factor to zero and solve the linear equations.