15. Quadratics and factoring
Vieta's: sum = -b/a, product = c/a. Factoring by grouping.
Core ideas
- Vieta's for x^2 + bx + c = 0 (a = 1): sum of roots = -b, product of roots = c. General ax^2 + bx + c: sum = -b/a, product = c/a.
- If a question asks about roots but not their exact values, reach for sum and product before solving.
- Factoring by grouping: split four terms into two pairs, pull the common factor from each, then factor out the shared binomial.
- Difference of squares a^2 - b^2 = (a - b)(a + b) shows up constantly after grouping.
Worked example 1
The equation x^2 - 7x + k = 0 has two roots that differ by 1. What is k?
Show solution
By Vieta's, the roots sum to 7 and their product is k. Two numbers that sum to 7 and differ by 1 are 3 and 4. So k = 3 * 4 = 12. Check: x^2 - 7x + 12 = (x - 3)(x - 4), roots 3 and 4, which differ by 1. No quadratic formula needed.
Worked example 2
Solve x^3 + 2x^2 - 9x - 18 = 0.
Show solution
Group: (x^3 + 2x^2) + (-9x - 18) = x^2(x + 2) - 9(x + 2). Factor out the shared binomial: (x + 2)(x^2 - 9). The second factor is a difference of squares: (x + 2)(x - 3)(x + 3) = 0. So x = -2, x = 3, or x = -3.
Practice set
Question 1
What is the sum of all roots of the equation x^3 + 3x^2 - 4x - 12 = 0?
Group the first two terms and the last two terms to pull out a common factor.
Factoring by grouping: x^3 + 3x^2 - 4x - 12 = x^2(x + 3) - 4(x + 3) = (x + 3)(x^2 - 4) = (x + 3)(x - 2)(x + 2). The roots are -3, 2, and -2, which sum to -3. This matches Vieta's shortcut for a cubic: the sum of the roots is minus the coefficient of x^2, here -3. Choosing 3 misses that sign flip.
Question 2
If r and s are the roots of x^2 - 6x + 4 = 0, what is the value of r^2 + s^2?
You never need the roots themselves; expand (r + s)^2 and rearrange.
By Vieta's formulas, r + s = 6 and rs = 4. Since (r + s)^2 = r^2 + 2rs + s^2, we get r^2 + s^2 = (r + s)^2 - 2rs = 36 - 8 = 28. The answer 36 forgets to subtract 2rs, and 32 subtracts rs only once.
Question 3
What is the sum of the roots of the equation x^2 - 7x + 12 = 0?
The coefficient of x already tells you the sum; watch the sign.
By Vieta's formulas, for x^2 + bx + c = 0 the sum of the roots is -b. Here b = -7, so the sum is 7. Factoring confirms it: (x - 3)(x - 4) = 0 gives roots 3 and 4, which sum to 7. Choosing -7 forgets the sign flip, and 12 is the product of the roots.
Question 4
The equation x^2 - kx + 16 = 0 has two equal roots. If k is positive, what is the value of k?
Equal roots force the quadratic to be a perfect square, or equivalently make the discriminant zero.
Equal roots require the discriminant to be zero: k^2 - 4(1)(16) = 0, so k^2 = 64 and k = 8 since k is positive. Equivalently, x^2 - 8x + 16 = (x - 4)^2. The answer 4 is the repeated root itself, not k.