Solve a quadratic exactly with the quadratic formula

Base-intersection equation

\(\displaystyle 2x^{2}-7x+2=0\)

A parabolic arch meets its base where 2x squared minus 7x plus 2 equals zero. Which exact expression gives both positions where the arch meets the base?

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

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

Exact roots from the quadratic formula

A quadratic equation in zero form has exact roots given by the quadratic formula. The denominator divides the whole numerator, and both signs are needed before a root is selected or rounded

1. 1Write zero form Move every term to one side so the equation reads ax squared plus bx plus c equals zero \(\displaystyle 2x^{2}-7x+2=0\)
2. 2Identify signed coefficients Read a, b and c with their signs attached \(\displaystyle a=2,\quad b=-7,\quad c=2\)
3. 3Calculate the discriminant Substitute the signed coefficients and simplify exactly \(\displaystyle \Delta=(-7)^{2}-4(2)(2)=33\)
4. 4Use both branches Keep the whole numerator over 2a and keep both signs for two real roots \(\displaystyle x=\frac{7\pm\sqrt{33}}{4}\)

Height model

\(\displaystyle h(t)=-2t^{2}+8t+3\)

Domain

\(\displaystyle t\ge 0\)

Ground condition

\(\displaystyle h=0\)

A firework's height is h(t) = -2t squared plus 8t plus 3 metres, with t at least zero. It reaches the ground when h equals zero. Find the exact ground time in seconds

Write the zero form, use the formula with both signs, then keep the root that satisfies the domain

Height model

\(\displaystyle h(t)=-2t^{2}+8t+3\)

Domain

\(\displaystyle t\ge 0\)

Ground condition

\(\displaystyle h=0\)

Using the same firework flight, find that ground time to the nearest 0.1 second

Approximate the selected exact root first, then round that value to one decimal place

For the equation shown, which line correctly gives the positive-sign branch of the quadratic formula?