Thirty-Six

Unit 4 · Expressions, Quadratics & PolynomialsA5 · Quadratics · Structures

Solving Quadratics

Choose factoring, square roots, the quadratic formula, or Desmos for any quadratic.

  • 30 min
  • Unlocks: Parabolic Bulkheads
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

discriminant
b2−4acb^2-4ac, which predicts the number of real roots

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

Factor when integers are visible; otherwise use the quadratic formula or complete the square. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Factor when integers are visible; otherwise use the quadratic formula or complete the square. Keep exact values through the setup; round only when the stem asks.
Core relationship
x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Factoring is fastest for integer roots, while the quadratic formula preserves irrational roots exactly. Desmos is efficient for checking decimal root locations by graphing the equation and clicking each solution.

Explore in Desmos

Read the roots of two quadratics

Type each equation as written and click every vertical solution line.

The calculator is live: change anything and see what happens.

Read the roots of two quadratics

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Trap

The main trap

Taking a square root requires both the positive and negative cases.

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Solve x2−7x+12=0x^2-7x+12=0.
Find numbers that multiply to 12 and add to −7-7: −3,−4-3,-4.

After factoring, what solutions follow from (x−3)(x−4)=0(x-3)(x-4)=0?

The solutions are x=3x=3 and x=4x=4.

Why may each factor be set equal to zero?

Worked example

Example 2: ACT twist

Problem

Solve 2x2+3x−4=02x^2+3x-4=0 exactly.
Here factoring is not quick, so use a=2,b=3,c=−4a=2,b=3,c=-4.

Which calculation gives the discriminant b2−4acb^2-4ac?

The solutions are x=−3±414x=\frac{-3\pm\sqrt{41}}4.

Why is the value under the radical positive 41?

Faded example · you finish it

Example 3: You finish it

Problem

Solve (x−5)2=16(x-5)^2=16.
x−5=±4x-5=\pm4

Complete the calculation.

Desmos way

Graphical root check

Type each equation as written and click every vertical solution line.

Graphical root check

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.