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Solve a quadratic exactly with the quadratic formula

Start a set of 8
Step 1 of 6
Base-intersection equation
\(\displaystyle 2x^{2}-7x+2=0\)

A parabolic arch meets its base at the roots of the equation shown. Which exact expression gives both positions?

Year 11, VCE Units 1 & 2, Chapter 3, 4 difficulty levels, Calculator allowedBuild a worksheet

The quadratic formula

For a quadratic in zero form with real coefficients and a nonzero leading coefficient, calculate the discriminant. A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives no real roots. Divide the whole numerator by twice the leading coefficient and simplify before selecting or rounding a root

\[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\]
  1. Write zero form Move every term to one side to obtain zero form. Check that the leading coefficient is nonzero before using the quadratic formula \(\displaystyle 2x^{2}-5x+2=0\)
  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 the discriminant and simplify exactly \(\displaystyle \Delta=b^{2}-4ac=(-5)^{2}-4(2)(2)=9\)
  4. Use both branches For a positive discriminant, keep both signs to obtain two distinct real roots. At zero the branches coincide. Divide the whole numerator by twice the leading coefficient \(\displaystyle x=\frac{-b\pm\sqrt{\Delta}}{2a}\)

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
✕\(\displaystyle x=\frac{-b+\sqrt{\Delta}}{2a}\)✓\(\displaystyle x=\frac{-b\pm\sqrt{\Delta}}{2a}\)
When Δ>0, the branches are distinct, so keeping only one omits a real root. When Δ=0, the branches coincide and the repeated root is recorded once

Difficulty levels

introInteger coefficients with two distinct rational roots
standardAn irreducible positive discriminant requiring an exact surd pair
challengeRational coefficients, including a repeated root, or one irrational branch rounded only after finding the exact roots
examCoefficient identification followed by an exact quadratic-formula result

Questions about this topic

Why must the equation be in zero form?+

The coefficients are read from a quadratic in zero form, so all terms must be on one side before they are identified

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 square root symbol gives the non-negative principal square root. For a positive discriminant, solving the squared equation gives both a positive and a negative value, 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, Mathematical Methods, chapter 3: Quadratic functions

In this chapter

Factorise, solve and reconstruct a rational quadraticCalculate and interpret a quadratic discriminantTransform a quadratic and find its key featuresComplete the square and link equivalent formsDetermine a quadratic rule from exact informationSolve a quadratic exactly with the quadratic formulaConstruct a rational quadratic from a surd rootSolve an equation that reduces to quadratic formAnalyse signs and solve a quadratic inequalityIntersections of a line and a quadratic graphDetermine parameter values from a discriminant conditionFormulate and solve a quadratic problemAnalyse and optimise a quadratic model