How to Solve Quadratic Equations

How to Solve Quadratic Equations

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Using the Quadratic Formula to solve quadratic equations

To use the quadratic formula, we must change the quadratic equation we are solving into "standard form". If we don't do this, none of the following steps will work. The goal is to change the quadratic equation so that the quadratic expression is on one side of the equation and only the number zero, 0, is on the other. Look at the equation below.

ax2 + bx + c = 0

This format makes it easy to find out what a, b, and c's numerical values are! Once we know these values, we can plug them into the quadratic formula and figure out the values of x.

The formula for quadratic equations

ax2 + bx + c = 0  x = b ± b2  4ac2a

Where a, b, and c are the coefficients of any quadratic equation in the standard form,  ax2 + bx + c = 0.

Simplify Be careful at each step to make the expressions simpler. Students often make mistakes because they tend to "chill out," which can lead to errors that could have been avoided, like when adding, subtracting, multiplying, or dividing real numbers.

Example:

Solve 2x2 + 3x  2 = 0.

Solution:

Given,

2x2 + 3x  2 = 0  a = 2, b = 3, c = 2

We know that b2  4ac = 32  4(2)(2) = 9  (16) = 25 > 0.

So, the roots exist. So, x = b ± b2  4ac2a = 3 ± 254

i.e., x1 = 3  54 = 84 = 2 and x2 = 3 + 54 = 24 = 12

So, the answer to 2x2 + 3x  2 = 0 is either 2 or 12 .

Solving Quadratic Equations By Factorisation

The quadratic equation can be written as the product of factors whose degree is less than or equal to two. It's called "factoring the quadratic equation," a way to solve quadratic equations.

Let's see an example to learn.

Example:

Use the factoring method to solve 2x2 + 6x + 4 =0.

Solution:

2x2 + 6x + 4 =0  2x2 + 2x + 4x + 4 = 0  2x(x + 1) + 4(x + 1) =0  (x + 1)(2x + 4) =0  x + 1 =0 or 2x + 4 =0 :

  • x + 1 =0  x = 1
  • 2x + 4 =0  2x = 4  x = 2

So, the answer to the given equation is either x = 1 or x = 2.

Free printable Worksheets

Exercises for Solving Quadratic Equations

1) Solve: x2 + 9x + 20 = 0

2) Solve: x2 + 2x  3 = 0

3) Solve: 2x2 + 7x  4 = 0

4) Solve: 2x2 + 4x + 2 = 0

5) Solve: 18x2  13x + 2 = 0

6) Solve: 5x2 + 27x  18 = 0

7) Solve: 2x2 + 7x + 3 = 0

8) Solve: 6x2 + 29x  35 = 0

9) Solve: 4x2 + 8x + 3 = 0

10) Solve: x2 + 3x  4 = 0

 

1) Solve: x2 + 9x + 20 = 0

x2 + 2x  3 = 0  (x + 4)(x + 5) = 0

  • (x + 4) = 0  x = 4
  • (x + 5) = 0  x = 5

x1 = 4 , x2 = 5

2) Solve: x2 + 2x  3 = 0

x2 + 9x + 20 = 0  a = 1, b = 2, c = 3
Δ = b2  4ac = 22  4(1)(3) = 4  (12) = 16
x = b ± Δ2a = 2 ± 162  x1 = 2  42 = 62 = 3 , x2 = 2 + 42 = 22 = 1

3) Solve: 2x2 + 7x  4 = 0

x2 + 9x + 20 = 0  a = 2, b = 7, c = 4
Δ = b2  4ac = 72  4(2)(4) = 49  (32) = 81
x = b ± Δ2a = 7 ± 814  x1 = 7  94 = 164 = 4 , x2 = 7 + 94 = 24 = 12

4) Solve: 2x2 + 4x + 2 = 0

2x2 + 4x + 2 = 0  2x2 + 2x + 2x + 2 = 0  2x(x + 1) + 2(x + 1) = 0  (x + 1)(2x + 2) = 0

  • (x + 1) = 0  x = 1
  • (2x + 2) = 0  x = 1

x1 = x2 = 1

5) Solve: 18x2  13x + 2 = 0

18x2  13x + 2 = 0  a = 18, b = 13, c = 2
Δ = b2  4ac = (13)2  4(18)(2) = 169  144 = 25
x = b ± Δ2a = 13 ± 2536  x1 = 13  536 = 836 = 29 ,  x2 = 13 + 536 = 1836 = 12

6) Solve: 5x2 + 27x  18 = 0

5x2 + 27x  18 = 0  a = 5, b = 27, c = 18
Δ = b2  4ac = (27)2  4(5)(18) = 729  (360) = 1089
x = b ± Δ2a = 27 ± 108910  x1 = 27  3310 = 6010 = 6 , x2 = 27 + 3310 = 610 = 0.6

7) Solve: 2x2 + 7x + 3 = 0

2x2 + 7x + 3 = 0  2x2 + 6x + x + 3 = 0  2x(x + 3) + 1(x + 3) = 0  (2x + 1)(x + 3) = 0

  • (2x + 1) = 0  x = 12
  • (x + 3) = 0  x = 3

x1 = 12 , x2 = 3

8) Solve: 6x2 + 29x  35 = 0

6x2 + 29x  35 = 0  6x2 + 14x + 15x  35 = 0   2x(3x  7) + 5(3x  7) = 0 (2x + 5)(3x  7) = 0

  • (2x + 5) = 0  x = 52 = 2.5
  • (3x  7) = 0  x = 73

x1 = 2.5 , x2 = 73

9) Solve: 4x2 + 8x + 3 = 0

4x2 + 8x + 3 = 0  a = 4, b = 8, c = 3
Δ = b2  4ac = (8)2  4(4)(3) = 64  48 = 16
x = b ± Δ2a = 8 ± 168  x1 = 8  48 = 128 = 32 ,  x2 = 8 + 48 = 48 = 12

10) Solve: x2 + 3x  4 = 0

x2 + 3x  4 = 0  a = 1, b = 3, c = 4
Δ = b2  4ac = (3)2  4(1)(4) = 9 + 16 = 25
x = b ± Δ2a = 3 ± 252  x1 = 3  52 = 92 = 4.5 ,  x2 = 3 + 52 = 22 = 1

Solving Quadratic Equations Practice Quiz