Mastering Quadratic Equations
What you will learn
A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. The standard form is \( ax^2 + bx + c = 0 \). Below are 25 fully solved examples to help you understand factorization and formula methods, followed by 10 practice problems to test your skills.
Note: Try to solve the equation yourself first, then click "Show Solution" to verify your method.
25 Solved Examples
01 Solve: \( x^2 + 5x + 6 = 0 \)
View Step-by-Step Solution
(i) Identify coefficients
Here, \(a=1, b=5, c=6\). We need two numbers that multiply to 6 and add to 5.
(ii) Factorize The numbers are 2 and 3. \[ x^2 + 2x + 3x + 6 = 0 \] \[ x(x + 2) + 3(x + 2) = 0 \] \[ (x + 2)(x + 3) = 0 \]
(iii) Find Roots Either \(x+2=0\) or \(x+3=0\).
Answer: \( x = -2, x = -3 \)
(ii) Factorize The numbers are 2 and 3. \[ x^2 + 2x + 3x + 6 = 0 \] \[ x(x + 2) + 3(x + 2) = 0 \] \[ (x + 2)(x + 3) = 0 \]
(iii) Find Roots Either \(x+2=0\) or \(x+3=0\).
Answer: \( x = -2, x = -3 \)
02 Solve: \( x^2 - 5x + 6 = 0 \)
View Step-by-Step Solution
(i) Find factors
We need numbers that multiply to \(+6\) and add to \(-5\). These are \(-2\) and \(-3\).
(ii) Split the middle term \[ x^2 - 2x - 3x + 6 = 0 \] \[ x(x - 2) - 3(x - 2) = 0 \] \[ (x - 2)(x - 3) = 0 \]
(iii) Solve Answer: \( x = 2, x = 3 \)
(ii) Split the middle term \[ x^2 - 2x - 3x + 6 = 0 \] \[ x(x - 2) - 3(x - 2) = 0 \] \[ (x - 2)(x - 3) = 0 \]
(iii) Solve Answer: \( x = 2, x = 3 \)
03 Solve: \( x^2 - 9 = 0 \)
View Step-by-Step Solution
(i) Use identity
Recall \( a^2 - b^2 = (a+b)(a-b) \). Here \( 9 = 3^2 \).
(ii) Factorize \[ x^2 - 3^2 = 0 \] \[ (x + 3)(x - 3) = 0 \]
(iii) Solve Answer: \( x = -3, x = 3 \)
(ii) Factorize \[ x^2 - 3^2 = 0 \] \[ (x + 3)(x - 3) = 0 \]
(iii) Solve Answer: \( x = -3, x = 3 \)
04 Solve: \( x^2 + 7x + 12 = 0 \)
View Step-by-Step Solution
(i) Find factors
Product = 12, Sum = 7. Factors are 3 and 4.
(ii) Factorize \[ (x + 3)(x + 4) = 0 \]
(iii) Solve Answer: \( x = -3, x = -4 \)
(ii) Factorize \[ (x + 3)(x + 4) = 0 \]
(iii) Solve Answer: \( x = -3, x = -4 \)
05 Solve: \( 2x^2 + 3x + 1 = 0 \)
View Step-by-Step Solution
(i) AC Method
Multiply \(a \times c = 2 \times 1 = 2\). We need sum = 3. Factors are 2 and 1.
(ii) Split middle term \[ 2x^2 + 2x + x + 1 = 0 \] \[ 2x(x + 1) + 1(x + 1) = 0 \] \[ (2x + 1)(x + 1) = 0 \]
(iii) Solve \( 2x = -1 \rightarrow x = -1/2 \) or \( x = -1 \).
Answer: \( x = -\frac{1}{2}, x = -1 \)
(ii) Split middle term \[ 2x^2 + 2x + x + 1 = 0 \] \[ 2x(x + 1) + 1(x + 1) = 0 \] \[ (2x + 1)(x + 1) = 0 \]
(iii) Solve \( 2x = -1 \rightarrow x = -1/2 \) or \( x = -1 \).
Answer: \( x = -\frac{1}{2}, x = -1 \)
06 Solve: \( x^2 - 2x - 15 = 0 \)
View Step-by-Step Solution
(i) Factors
Product = -15, Sum = -2. Factors: -5 and +3.
(ii) Factorize \[ (x - 5)(x + 3) = 0 \]
(iii) Solve Answer: \( x = 5, x = -3 \)
(ii) Factorize \[ (x - 5)(x + 3) = 0 \]
(iii) Solve Answer: \( x = 5, x = -3 \)
07 Solve: \( x^2 - 8x + 16 = 0 \)
View Step-by-Step Solution
(i) Analyze structure
This is a perfect square trinomial because \( (-4)^2 = 16 \) and \( 2(-4) = -8 \).
(ii) Factorize \[ (x - 4)^2 = 0 \]
(iii) Solve Answer: \( x = 4 \) (Equal real roots)
(ii) Factorize \[ (x - 4)^2 = 0 \]
(iii) Solve Answer: \( x = 4 \) (Equal real roots)
08 Solve: \( 3x^2 - 5x + 2 = 0 \)
View Step-by-Step Solution
(i) AC Method
\( a \times c = 3 \times 2 = 6 \). Sum = -5. Factors: -3 and -2.
(ii) Split middle term \[ 3x^2 - 3x - 2x + 2 = 0 \] \[ 3x(x - 1) - 2(x - 1) = 0 \] \[ (3x - 2)(x - 1) = 0 \]
(iii) Solve Answer: \( x = \frac{2}{3}, x = 1 \)
(ii) Split middle term \[ 3x^2 - 3x - 2x + 2 = 0 \] \[ 3x(x - 1) - 2(x - 1) = 0 \] \[ (3x - 2)(x - 1) = 0 \]
(iii) Solve Answer: \( x = \frac{2}{3}, x = 1 \)
09 Solve: \( x^2 + 4x = 0 \)
View Step-by-Step Solution
(i) Take common factor
There is no constant term \(c\). Take \(x\) common.
(ii) Factorize \[ x(x + 4) = 0 \]
(iii) Solve Answer: \( x = 0, x = -4 \)
(ii) Factorize \[ x(x + 4) = 0 \]
(iii) Solve Answer: \( x = 0, x = -4 \)
10 Solve: \( 2x^2 - 7x + 3 = 0 \)
View Step-by-Step Solution
(i) AC Method
Product = 6, Sum = -7. Factors: -6 and -1.
(ii) Split middle term \[ 2x^2 - 6x - x + 3 = 0 \] \[ 2x(x - 3) - 1(x - 3) = 0 \] \[ (2x - 1)(x - 3) = 0 \]
(iii) Solve Answer: \( x = \frac{1}{2}, x = 3 \)
(ii) Split middle term \[ 2x^2 - 6x - x + 3 = 0 \] \[ 2x(x - 3) - 1(x - 3) = 0 \] \[ (2x - 1)(x - 3) = 0 \]
(iii) Solve Answer: \( x = \frac{1}{2}, x = 3 \)
11 Solve: \( x^2 - 4x - 21 = 0 \)
View Step-by-Step Solution
(i) Factors
Product = -21, Sum = -4. Factors: -7 and +3.
(ii) Factorize \[ (x - 7)(x + 3) = 0 \]
(iii) Solve Answer: \( x = 7, x = -3 \)
(ii) Factorize \[ (x - 7)(x + 3) = 0 \]
(iii) Solve Answer: \( x = 7, x = -3 \)
12 Solve using Formula: \( x^2 + 4x + 2 = 0 \)
View Step-by-Step Solution
(i) Quadratic Formula
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \(a=1, b=4, c=2\).
(ii) Substitute \[ x = \frac{-4 \pm \sqrt{16 - 8}}{2} \] \[ x = \frac{-4 \pm \sqrt{8}}{2} \] \[ x = \frac{-4 \pm 2\sqrt{2}}{2} \]
(iii) Simplify Answer: \( x = -2 \pm \sqrt{2} \)
(ii) Substitute \[ x = \frac{-4 \pm \sqrt{16 - 8}}{2} \] \[ x = \frac{-4 \pm \sqrt{8}}{2} \] \[ x = \frac{-4 \pm 2\sqrt{2}}{2} \]
(iii) Simplify Answer: \( x = -2 \pm \sqrt{2} \)
13 Solve: \( 6x^2 - x - 2 = 0 \)
View Step-by-Step Solution
(i) AC Method
Product = -12, Sum = -1. Factors: -4 and +3.
(ii) Split middle term \[ 6x^2 - 4x + 3x - 2 = 0 \] \[ 2x(3x - 2) + 1(3x - 2) = 0 \] \[ (2x + 1)(3x - 2) = 0 \]
(iii) Solve Answer: \( x = -\frac{1}{2}, x = \frac{2}{3} \)
(ii) Split middle term \[ 6x^2 - 4x + 3x - 2 = 0 \] \[ 2x(3x - 2) + 1(3x - 2) = 0 \] \[ (2x + 1)(3x - 2) = 0 \]
(iii) Solve Answer: \( x = -\frac{1}{2}, x = \frac{2}{3} \)
14 Solve: \( 4x^2 - 12x + 9 = 0 \)
View Step-by-Step Solution
(i) Identify square
\( (2x)^2 - 2(2x)(3) + 3^2 = 0 \).
(ii) Factorize \[ (2x - 3)^2 = 0 \]
(iii) Solve Answer: \( x = \frac{3}{2} \) (Repeated root)
(ii) Factorize \[ (2x - 3)^2 = 0 \]
(iii) Solve Answer: \( x = \frac{3}{2} \) (Repeated root)
15 Solve: \( x^2 + x - 2 = 0 \)
View Step-by-Step Solution
(i) Factors
Product = -2, Sum = +1. Factors: +2 and -1.
(ii) Factorize \[ (x + 2)(x - 1) = 0 \]
(iii) Solve Answer: \( x = -2, x = 1 \)
(ii) Factorize \[ (x + 2)(x - 1) = 0 \]
(iii) Solve Answer: \( x = -2, x = 1 \)
16 Solve: \( 5x^2 = 20 \)
View Step-by-Step Solution
(i) Isolate x squared
Divide by 5: \( x^2 = 4 \).
(ii) Square root \( x = \pm\sqrt{4} \).
(iii) Solve Answer: \( x = 2, x = -2 \)
(ii) Square root \( x = \pm\sqrt{4} \).
(iii) Solve Answer: \( x = 2, x = -2 \)
17 Solve: \( x^2 - 11x + 24 = 0 \)
View Step-by-Step Solution
(i) Factors
Product = 24, Sum = -11. Factors: -8 and -3.
(ii) Factorize \[ (x - 8)(x - 3) = 0 \]
(iii) Solve Answer: \( x = 8, x = 3 \)
(ii) Factorize \[ (x - 8)(x - 3) = 0 \]
(iii) Solve Answer: \( x = 8, x = 3 \)
18 Solve: \( x^2 + 10x + 25 = 0 \)
View Step-by-Step Solution
(i) Perfect Square
This fits \( (a+b)^2 \).
(ii) Factorize \[ (x + 5)^2 = 0 \]
(iii) Solve Answer: \( x = -5 \)
(ii) Factorize \[ (x + 5)^2 = 0 \]
(iii) Solve Answer: \( x = -5 \)
19 Solve: \( 2x^2 + 5x - 3 = 0 \)
View Step-by-Step Solution
(i) AC Method
Product = -6, Sum = 5. Factors: +6 and -1.
(ii) Split middle term \[ 2x^2 + 6x - x - 3 = 0 \] \[ 2x(x + 3) - 1(x + 3) = 0 \] \[ (2x - 1)(x + 3) = 0 \]
(iii) Solve Answer: \( x = \frac{1}{2}, x = -3 \)
(ii) Split middle term \[ 2x^2 + 6x - x - 3 = 0 \] \[ 2x(x + 3) - 1(x + 3) = 0 \] \[ (2x - 1)(x + 3) = 0 \]
(iii) Solve Answer: \( x = \frac{1}{2}, x = -3 \)
20 Solve using Formula: \( x^2 - 6x + 7 = 0 \)
View Step-by-Step Solution
(i) Apply Formula
\( x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(7)}}{2(1)} \).
(ii) Calculate \[ x = \frac{6 \pm \sqrt{36 - 28}}{2} \] \[ x = \frac{6 \pm \sqrt{8}}{2} \] \[ x = \frac{6 \pm 2\sqrt{2}}{2} \]
(iii) Simplify Answer: \( x = 3 \pm \sqrt{2} \)
(ii) Calculate \[ x = \frac{6 \pm \sqrt{36 - 28}}{2} \] \[ x = \frac{6 \pm \sqrt{8}}{2} \] \[ x = \frac{6 \pm 2\sqrt{2}}{2} \]
(iii) Simplify Answer: \( x = 3 \pm \sqrt{2} \)
21 Solve: \( x^2 + 3x - 10 = 0 \)
View Step-by-Step Solution
(i) Factors
Product = -10, Sum = 3. Factors: +5 and -2.
(ii) Factorize \[ (x + 5)(x - 2) = 0 \]
(iii) Solve Answer: \( x = -5, x = 2 \)
(ii) Factorize \[ (x + 5)(x - 2) = 0 \]
(iii) Solve Answer: \( x = -5, x = 2 \)
22 Solve: \( 3x^2 = 2x \)
View Step-by-Step Solution
(i) Rearrange
\[ 3x^2 - 2x = 0 \]
(ii) Factorize common term \[ x(3x - 2) = 0 \]
(iii) Solve \( x = 0 \) or \( 3x - 2 = 0 \).
Answer: \( x = 0, x = \frac{2}{3} \)
(ii) Factorize common term \[ x(3x - 2) = 0 \]
(iii) Solve \( x = 0 \) or \( 3x - 2 = 0 \).
Answer: \( x = 0, x = \frac{2}{3} \)
23 Solve: \( x^2 - 13x + 42 = 0 \)
View Step-by-Step Solution
(i) Factors
Product = 42, Sum = -13. Factors: -6 and -7.
(ii) Factorize \[ (x - 6)(x - 7) = 0 \]
(iii) Solve Answer: \( x = 6, x = 7 \)
(ii) Factorize \[ (x - 6)(x - 7) = 0 \]
(iii) Solve Answer: \( x = 6, x = 7 \)
24 Solve: \( 4x^2 - 1 = 0 \)
View Step-by-Step Solution
(i) Difference of Squares
\( (2x)^2 - 1^2 = 0 \).
(ii) Factorize \[ (2x - 1)(2x + 1) = 0 \]
(iii) Solve Answer: \( x = \frac{1}{2}, x = -\frac{1}{2} \)
(ii) Factorize \[ (2x - 1)(2x + 1) = 0 \]
(iii) Solve Answer: \( x = \frac{1}{2}, x = -\frac{1}{2} \)
25 Solve: \( x^2 + 8x + 15 = 0 \)
View Step-by-Step Solution
(i) Factors
Product = 15, Sum = 8. Factors: 3 and 5.
(ii) Factorize \[ (x + 3)(x + 5) = 0 \]
(iii) Solve Answer: \( x = -3, x = -5 \)
(ii) Factorize \[ (x + 3)(x + 5) = 0 \]
(iii) Solve Answer: \( x = -3, x = -5 \)
Practice Questions
Try solving these 10 questions on your own before checking the answer key below.
(i) \( x^2 + 7x + 10 = 0 \)
(ii) \( x^2 - 3x - 10 = 0 \)
(iii) \( 2x^2 + 5x + 3 = 0 \)
(iv) \( x^2 - 49 = 0 \)
(v) \( x^2 - 6x = 0 \)
(vi) \( x^2 + 12x + 36 = 0 \)
(vii) \( 3x^2 - x - 4 = 0 \)
(viii) \( x^2 - x - 30 = 0 \)
(ix) \( x^2 - 10x + 21 = 0 \)
(x) \( 2x^2 + 7x - 4 = 0 \)
Check Answer Key
(i) \( x = -2, x = -5 \)
(ii) \( x = 5, x = -2 \)
(iii) \( x = -1, x = -3/2 \)
(iv) \( x = 7, x = -7 \)
(v) \( x = 0, x = 6 \)
(vi) \( x = -6 \)
(vii) \( x = -1, x = 4/3 \)
(viii) \( x = 6, x = -5 \)
(ix) \( x = 3, x = 7 \)
(x) \( x = 1/2, x = -4 \)