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Solve a quadratic exactly with the quadratic formula

Start a set of 8
Step 2 of 6

Start with the discriminant at 0, then move it up step by step. Watch the two root positions on the line and the four readouts below it

Number line from 1 to 5 with two points that start together at 3 and move apart as the discriminant increases11.522.533.544.55Root position
Lower branch
3
Upper branch
3
Midpoint
3
Distance between roots
0
0

Across all positions of the slider, what happened as the discriminant increased from 0?

Year 11, VCE Units 1 & 2, Chapter 9, 4 difficulty levels, Calculator allowedBuild a worksheet

The quadratic formula

The quadratic formula gives the exact roots of ax² + bx + c = 0. Calculate the discriminant first, then substitute the signed coefficients and simplify both branches

\[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\]
  1. Write zero form Use the form ax² + bx + c = 0 before reading the three coefficients \(\displaystyle 2x^{2}-5x+2=0\)
  2. Identify signed coefficients Keep the sign attached to b and c when reading the equation \(\displaystyle a=2,\quad b=-5,\quad c=2\)
  3. Calculate the discriminant Substitute the signed coefficients into b² − 4ac and simplify exactly \(\displaystyle \Delta=b^{2}-4ac=(-5)^{2}-4(2)(2)=9\)
  4. Use both branches Keep the whole numerator over 2a and use both signs when the roots are real \(\displaystyle x=\frac{-b\pm\sqrt{\Delta}}{2a}\)

Worked example

Solve the quadratic equation exactly

\[x^{2}-7x+10=0\]
  1. \(\displaystyle a=1,\quad b=-7,\quad c=10\)Read the signed coefficients from the equation in zero form
  2. \(\displaystyle \Delta=(-7)^{2}-4(1)(10)=49-40=9\)Square the signed value of b, then subtract 4ac
  3. \(\displaystyle x=\frac{-(-7)\pm\sqrt{9}}{2(1)}=\frac{7\pm3}{2}\)Substitute the coefficients with the whole numerator over 2a
  4. \(\displaystyle x=2\quad\text{or}\quad x=5\)The two branches give (7 − 3) ÷ 2 = 2 and (7 + 3) ÷ 2 = 5

Common mistakes

Dividing only part of the numerator
✕\(\displaystyle x=-b+\frac{\sqrt{b^{2}-4ac}}{2a}\)✓\(\displaystyle x=\frac{-b+\sqrt{b^{2}-4ac}}{2a}\)
The denominator 2a divides the whole numerator, so -b and the radical stay together above it
Forgetting to square b
✕\(\displaystyle \Delta=b-4ac\)✓\(\displaystyle \Delta=b^{2}-4ac\)
The discriminant uses the square of the signed linear coefficient, so b must be squared before 4ac is subtracted
Losing one branch
✕\(\displaystyle x=\frac{-b+\sqrt{\Delta}}{2a}\)✓\(\displaystyle x=\frac{-b\pm\sqrt{\Delta}}{2a}\)
The plus-minus sign gives the two possible branches, so both signs are needed for two real roots

Difficulty levels

introInteger coefficients with two distinct rational roots
standardAn irreducible positive discriminant requiring an exact surd pair
challengeRational coefficients or one exact root rounded only at the final step
examCoefficient identification followed by an exact quadratic-formula result

Questions about this topic

Why must the equation be in zero form?+

The coefficients a, b and c are defined by ax² + bx + c = 0, so all terms must be on one side before they are read

What does the discriminant tell you?+

A negative discriminant gives no real roots, zero gives one repeated real root, and a positive discriminant gives two real roots

Why does the formula contain plus-minus?+

The symbol √Δ denotes the non-negative principal square root. When Δ is positive, solving z² = Δ gives z = √Δ or z = −√Δ, so the plus-minus sign retains both branches

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 9: Complex numbers

In this chapter

Calculate with complex numbers in Cartesian formSolve a quadratic exactly with the quadratic formulaUse polar forms, arguments and complex multiplication geometryTranslate complex relations into loci and combined regions