Quadratics
Roots, the discriminant, the vertex, and the formula.
Pen and paper is fine · no calculator needed why?
Opens at level 28.
the lesson
Read the lesson
The idea
A quadratic is ax² + bx + c. Its graph is a curve called a parabola, shaped like a U or an upside-down U. Its roots are the values of x that make it 0, where the curve meets the x-axis.
The discriminant, b² − 4ac, tells you how many real roots there are: two if it is positive, one if it is 0, none if it is negative. The vertex, the turning point, sits where x is −b ÷ (2a). When the roots are not whole numbers, the quadratic formula finds them.
Techniques
Factor, then read the roots
- Divide out any number every term shares, like 2 or 3, so x² stands alone. Never divide by x.
- Find two numbers that multiply to c and add to b.
- Each bracket gives a root with the opposite sign: x + 3 gives −3, and x alone gives 0.
worked example
Solve 2x² + 2x − 12 = 0. What is the smaller root?
- Divide by 2: x² + x − 6 = 0.
- 3 and −2 multiply to −6 and add to 1.
- (x + 3)(x − 2) = 0, so x is −3 or 2.
- The smaller root is −3.
Answer: −3
Vertex from −b ÷ (2a)
- The x-coordinate is −b ÷ (2a). Keep the signs of both a and b.
- Put that x back into the equation to get y.
worked example
What is the y-coordinate of the vertex of y = 2x² − 8x + 3?
- x = −b ÷ (2a) = 8 ÷ 4 = 2.
- y = 2 × 2² − 8 × 2 + 3 = 8 − 16 + 3 = −5.
Answer: −5
The quadratic formula
- Write down a, b and c, with their signs.
- Work out the discriminant, b² − 4ac, first.
- x = (−b ± its square root) ÷ (2a). With a positive, + gives the larger root.
- Round only at the end.
worked example
Solve x² − 2x − 4 = 0. Give the larger root, rounded to 2 decimal places.
- a = 1, b = −2, c = −4.
- b² − 4ac = 4 + 16 = 20.
- x = (2 + √20) ÷ 2, and √20 ≈ 4.4721.
- (2 + 4.4721) ÷ 2 ≈ 3.236, which rounds to 3.24.
Answer: 3.24
Tips by skill
- TipRoots by factoring: Divide out any shared number, never x. Find two numbers that multiply to c and add to b; each bracket's root has the opposite sign.
- TipDiscriminant: Work out b² − 4ac with brackets around negatives. A negative b squared is positive.
- TipVertex: The x-coordinate is −b ÷ (2a). Put it back into the equation to get y. Enter x first.
- TipQuadratic formula: x = (−b ± √(b² − 4ac)) ÷ (2a). Use + for the larger root and − for the smaller. Round only at the end.
Watch out for
- Squaring a negative b and keeping the minus. The square of −5 is 25.
- Adding 4ac instead of subtracting it in the discriminant.
- Dropping the minus in −b ÷ (2a), which puts the vertex on the wrong side.
- Dividing every term by x. That throws away the root 0.
skills · practice stats
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Roots by factoring not tried yet
worked example
Solve 3x² − 21x = 0. Enter both roots, smaller first.
Answer: (0, 7)
- Divide by 3: x² − 7x = 0.
- Factor: x(x − 7) = 0.
- So x = 7 or x = 0. Smaller first: 0, 7.
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Discriminant not tried yet
worked example
What is the discriminant of 2x² − 4x + 10?
Answer: -64
- b² − 4ac = (−4)² − 4 × 2 × 10 = 16 − 80 = −64.
- It is negative, so there are no real roots.
-
Vertex not tried yet
worked example
What is the vertex of y = x² + 10x + 9? Enter x, y.
Answer: (-5, -16)
- x = −b ÷ (2a) = −10 ÷ 2 = −5.
- y = (−5)² + 10 × (−5) + 9 = 25 − 50 + 9 = −16.
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Quadratic formula not tried yet
worked example
Solve x² + 2x − 7 = 0. Give the larger root, rounded to 2 decimal places.
Answer: 1.83
- x = (−b ± √(b² − 4ac)) ÷ (2a) = (−2 ± √(4 − (−28))) ÷ 2 = (−2 ± √32) ÷ 2.
- √32 ≈ 5.6569, so the larger root is (−2 + 5.6569) ÷ 2 ≈ 1.8284.
- Rounded to 2 decimal places: 1.83.
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