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The quadratic formula

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Solve x^2 - 2x - 4 = 0 using the quadratic formula.

Step 1 of 3 · Answer this

Match the equation to ax^2 + bx + c = 0. Include the sign: what is c?

Common mix-up

In this example, a student wrote x = (2 + sqrt(-12))/2.

The student read c as positive 4, computed the discriminant as -12, and stopped, never simplifying the radical or dividing the numerator.

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See why this works

What is x when x^2 - 2x - 4 = 0?

Not every quadratic factors over the integers. The quadratic formula works for all of them, and it only asks you for three careful numbers.

For ax^2 + bx + c = 0, x = (-b +/- sqrt(b^2 - 4ac)) / 2a.

The quadratic formula solves any equation of the form ax^2 + bx + c = 0 in one routine. For x^2 - 2x - 4 = 0, the matching numbers are a = 1, b = -2, and c = -4. The sign carries with each number: b is negative two, not two. Copying the signs wrongly is the most common first slip.

The discriminant inside the square root is b^2 - 4ac. With these values, that is (-2)^2 - 4(1)(-4) = 4 + 16 = 20. It comes out positive, so the equation has two real solutions and the formula will produce two of them, one for each sign of the plus or minus.

Dry-erase diagram showing quadratic formula steps for x^2 - 2x - 4 = 0, identifying a=1, b=-2, c=-4 and solving to 1 ± sqrt(5).
See the ideaIdentify coefficients with their signs (a=1, b=-2, c=-4), calculate the discriminant, and simplify to find x = 1 ± √5.

Now run the formula: x = (2 +/- sqrt(20)) / 2. Before dividing, simplify the radical: sqrt(20) = sqrt(4 * 5) = 2 * sqrt(5). Factor the numerator, divide out the 2 somewhere, and ends as x = 1 +/- sqrt(5). Simplifying the radical first keeps the division clean.

Check one solution by substituting. For x = 1 + sqrt(5), the square of x is 6 + 2*sqrt(5), then subtract 2x and 4: (6 + 2*sqrt(5)) - (2 + 2*sqrt(5)) - 4 = 0. The radical parts cancel and the whole expression collapses to zero, exactly as it should.

Keep this idea

Read a, b, c with their signs, work out b^2 - 4ac first, simplify the radical, then divide the whole numerator by 2a.

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