Quadratics Reference

Quadratic Formula

The discriminant is the part of the quadratic formula that tells us how many real x-intercepts a quadratic has and whether factoring is likely to work cleanly.

Solved Examples

Jump directly to a worked example by discriminant or equation type.

Fact Table

Result $D$ What It Means for the Graph Can You Factor It?
Positive square: $1,\ 4,\ 9,\ldots$ Hits the x-axis twice. Yes. Usually easy factoring.
Positive non-square: $2,\ 5,\ldots$ Hits the x-axis twice. No. Use the quadratic formula.
Zero: $0$ Hits the x-axis exactly once. Yes. Perfect square.
Negative: $-3,\ -10,\ldots$ Never hits the x-axis. No. Prime over the real numbers.

Content Formulas

Discriminant
$$D=b^2-4ac$$
Quadratic Formula
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Use the discriminant before solving when the question asks about x-intercepts, real solutions, or whether a quadratic can factor cleanly.

Classic Examples

Positive Square

For $x^2-3x+2=0$, decide what the graph does and solve.

Solution Steps
  1. Calculate the discriminant.
  2. Simplify the discriminant calculation.
  3. State that the positive square discriminant is $1$.
  4. Factor the quadratic.
  5. State the two real solutions: $x=1$ and $x=2$.
$$D=(-3)^2-4(1)(2)$$ $$D=9-8$$ $$D=1$$ $$(x-1)(x-2)=0$$ $$x=1,\quad x=2$$

Positive Non-Square

For $x^2-4x+2=0$, decide what the graph does and solve.

Solution Steps
  1. Calculate the discriminant.
  2. Simplify the discriminant calculation.
  3. State that the positive non-square discriminant is $8$.
  4. Substitute into the quadratic formula.
  5. State the two real solutions: $x=2\pm\sqrt2$.
$$D=(-4)^2-4(1)(1)$$ $$D=16-8$$ $$D=8$$ $$x=\frac{4\pm\sqrt{8}}{2}$$ $$x=2\pm\sqrt2$$

Zero

For $x^2-2x+1=0$, decide what the graph does and solve.

Solution Steps
  1. Calculate the discriminant.
  2. Simplify the discriminant calculation.
  3. State that the zero discriminant gives one real x-intercept.
  4. Rewrite the quadratic as a perfect square.
  5. State the repeated solution: $x=1$.
$$D=(-2)^2-4(1)(1)$$ $$D=4-4$$ $$D=0$$ $$(x-1)^2=0$$ $$x=1$$

Negative

For $x^2+2x+5=0$, decide what the graph does.

Solution Steps
  1. Calculate the discriminant.
  2. Simplify the discriminant calculation.
  3. State that the discriminant is negative.
  4. State the graph's result: no real x-intercepts.
$$D=2^2-4(1)(5)$$ $$D=4-20$$ $$D=-16$$ $$\text{No real x-intercepts.}$$

Extraneous Solution Check

Quadratic equations usually do not create extraneous solutions by themselves. Extraneous solutions appear when the original problem includes a square root, rational denominator, or another restriction.

Square Root Equation

Solve $\sqrt{x+5}=x-1$.

Solution Steps
  1. Square both sides of the original equation.
  2. Expand the squared binomial.
  3. Move all terms to one side.
  4. Factor the quadratic.
  5. State both candidate solutions.
  6. Test $x=-1$ in the original equation and reject it.
  7. State the valid solution: $x=4$.
$$x+5=(x-1)^2$$ $$x+5=x^2-2x+1$$ $$0=x^2-3x-4$$ $$(x-4)(x+1)=0$$ $$x=4,\quad x=-1$$ $$\sqrt{-1+5}\ne -1-1$$ $$x=4$$