Quadratic Equations

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Quick recall

• Check the sign of the discriminant before describing the roots.
• When splitting the middle term, the two numbers sum to b and multiply to ac.

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Formula Cards

1. General quadratic

$$ax^2+bx+c=0,\qquad a\ne0$$

2. Discriminant

$$D=b^2-4ac$$

3. Roots

$$x=\frac{-b\pm\sqrt{D}}{2a}$$

4. Nature of roots

D > 0: two distinct real roots. D = 0: a repeated real root. D < 0: two non-real complex roots.

5. Sum and product

$$\alpha+\beta=-\frac{b}{a},\qquad\alpha\beta=\frac{c}{a}$$

6. Split the middle term

$$p+q=b,\qquad pq=ac$$

Find p and q, replace bx by px+qx, then factor by grouping.

Worked Examples

Example 1

Question: Solve x²−5x+6 = 0.

Step 1: Find the discriminant.

$$D=(-5)^2-4(1)(6)=1$$

Step 2: Use the root formula.

$$x=\frac{5\pm1}{2}$$

Step 3: Evaluate both signs.

$$x=2\quad\text{or}\quad x=3$$

Answer: 2 and 3.

Example 2

Question: Solve x²−4x+4 = 0 and describe the roots.

Step 1: Calculate the discriminant.

$$D=(-4)^2-4(1)(4)=0$$

Step 2: Use the repeated-root formula.

$$x=\frac{4}{2}=2$$

Answer: A repeated real root: x = 2.

Example 3

Question: Find the roots of x²−7x+12 = 0 using sum and product.

Step 1: Identify the required sum and product.

$$\alpha+\beta=7,\qquad\alpha\beta=12$$

Step 2: Choose the pair and verify the factors.

$$3+4=7,\qquad3\times4=12$$

Step 3: Factor the polynomial.

$$(x-3)(x-4)=0$$

Answer: 3 and 4.

Drill

Try each question before opening its answer.

Question 1

Solve $x^2-3x-10=0$.

• Show answer

  x=5,-2

Question 2

Find nature of roots of $x^2+2x+5=0$.

• Show answer

  Complex (D=-16)

Question 3

Sum and product of roots of $2x^2-6x+4=0$.

• Show answer

  Sum=3, Product=2

Question 4

Solve $x^2-9=0$.

• Show answer

  x=3,-3

Question 5

Find k such that $x^2-kx+9=0$ has equal roots.

• Show answer

  k=6 or -6

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