Solve 5x^2-20x+12=x^2+1x-6

$$5 x ^ { 2 } - 20 x + 12 = x ^ { 2 } + 1 x - 6$$

Solve for x

$$x = \frac{3 \sqrt{17} + 21}{8} \approx 4.17116461$$

$$x = \frac{21 - 3 \sqrt{17}}{8} \approx 1.07883539$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$5x^{2}-20x+12-x^{2}=1x-6$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}-20x+12=1x-6$$

Subtract 1x from both sides.

$$4x^{2}-20x+12-x=-6$$

Combine -20x and -x to get -21x.

$$4x^{2}-21x+12=-6$$

Add 6 to both sides.

$$4x^{2}-21x+12+6=0$$

Add 12 and 6 to get 18.

$$4x^{2}-21x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -21 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 4\times 18}}{2\times 4}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 4\times 18}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-21\right)±\sqrt{441-16\times 18}}{2\times 4}$$

Multiply -16 times 18.

$$x=\frac{-\left(-21\right)±\sqrt{441-288}}{2\times 4}$$

Add 441 to -288.

$$x=\frac{-\left(-21\right)±\sqrt{153}}{2\times 4}$$

Take the square root of 153.

$$x=\frac{-\left(-21\right)±3\sqrt{17}}{2\times 4}$$

The opposite of -21 is 21.

$$x=\frac{21±3\sqrt{17}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{21±3\sqrt{17}}{8}$$

Now solve the equation x=\frac{21±3\sqrt{17}}{8} when ± is plus. Add 21 to 3\sqrt{17}.

$$x=\frac{3\sqrt{17}+21}{8}$$

Now solve the equation x=\frac{21±3\sqrt{17}}{8} when ± is minus. Subtract 3\sqrt{17} from 21.

$$x=\frac{21-3\sqrt{17}}{8}$$

The equation is now solved.

$$x=\frac{3\sqrt{17}+21}{8} x=\frac{21-3\sqrt{17}}{8}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$5x^{2}-20x+12-x^{2}=1x-6$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}-20x+12=1x-6$$

Subtract 1x from both sides.

$$4x^{2}-20x+12-x=-6$$

Combine -20x and -x to get -21x.

$$4x^{2}-21x+12=-6$$

Subtract 12 from both sides.

$$4x^{2}-21x=-6-12$$

Subtract 12 from -6 to get -18.

$$4x^{2}-21x=-18$$

Divide both sides by 4.

$$\frac{4x^{2}-21x}{4}=-\frac{18}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{21}{4}x=-\frac{18}{4}$$

Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{21}{4}x=-\frac{9}{2}$$

Divide -\frac{21}{4}, the coefficient of the x term, by 2 to get -\frac{21}{8}. Then add the square of -\frac{21}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{21}{4}x+\left(-\frac{21}{8}\right)^{2}=-\frac{9}{2}+\left(-\frac{21}{8}\right)^{2}$$

Square -\frac{21}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=-\frac{9}{2}+\frac{441}{64}$$

Add -\frac{9}{2} to \frac{441}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=\frac{153}{64}$$

Factor x^{2}-\frac{21}{4}x+\frac{441}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{8}\right)^{2}=\frac{153}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{8}\right)^{2}}=\sqrt{\frac{153}{64}}$$

Simplify.

$$x-\frac{21}{8}=\frac{3\sqrt{17}}{8} x-\frac{21}{8}=-\frac{3\sqrt{17}}{8}$$

Add \frac{21}{8} to both sides of the equation.

$$x=\frac{3\sqrt{17}+21}{8} x=\frac{21-3\sqrt{17}}{8}$$