Solve x(12x+7)-3x(20x-5)=-100

$$x ( 12 x + 7 ) - 3 x ( 20 x - 5 ) = - 100$$

Solve for x

$$x = \frac{\sqrt{4921} + 11}{48} \approx 1.690621659$$

$$x=\frac{11-\sqrt{4921}}{48}\approx -1.232288326$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 12x+7.

$$12x^{2}+7x-3x\left(20x-5\right)=-100$$

Add 100 to both sides.

$$12x^{2}+7x-3x\left(20x-5\right)+100=0$$

Use the distributive property to multiply -3x by 20x-5.

$$12x^{2}+7x-60x^{2}+15x+100=0$$

Combine 12x^{2} and -60x^{2} to get -48x^{2}.

$$-48x^{2}+7x+15x+100=0$$

Combine 7x and 15x to get 22x.

$$-48x^{2}+22x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -48 for a, 22 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\left(-48\right)\times 100}}{2\left(-48\right)}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\left(-48\right)\times 100}}{2\left(-48\right)}$$

Multiply -4 times -48.

$$x=\frac{-22±\sqrt{484+192\times 100}}{2\left(-48\right)}$$

Multiply 192 times 100.

$$x=\frac{-22±\sqrt{484+19200}}{2\left(-48\right)}$$

Add 484 to 19200.

$$x=\frac{-22±\sqrt{19684}}{2\left(-48\right)}$$

Take the square root of 19684.

$$x=\frac{-22±2\sqrt{4921}}{2\left(-48\right)}$$

Multiply 2 times -48.

$$x=\frac{-22±2\sqrt{4921}}{-96}$$

Now solve the equation x=\frac{-22±2\sqrt{4921}}{-96} when ± is plus. Add -22 to 2\sqrt{4921}.

$$x=\frac{2\sqrt{4921}-22}{-96}$$

Divide -22+2\sqrt{4921} by -96.

$$x=\frac{11-\sqrt{4921}}{48}$$

Now solve the equation x=\frac{-22±2\sqrt{4921}}{-96} when ± is minus. Subtract 2\sqrt{4921} from -22.

$$x=\frac{-2\sqrt{4921}-22}{-96}$$

Divide -22-2\sqrt{4921} by -96.

$$x=\frac{\sqrt{4921}+11}{48}$$

The equation is now solved.

$$x=\frac{11-\sqrt{4921}}{48} x=\frac{\sqrt{4921}+11}{48}$$

Steps for Completing the Square

Use the distributive property to multiply x by 12x+7.

$$12x^{2}+7x-3x\left(20x-5\right)=-100$$

Use the distributive property to multiply -3x by 20x-5.

$$12x^{2}+7x-60x^{2}+15x=-100$$

Combine 12x^{2} and -60x^{2} to get -48x^{2}.

$$-48x^{2}+7x+15x=-100$$

Combine 7x and 15x to get 22x.

$$-48x^{2}+22x=-100$$

Divide both sides by -48.

$$\frac{-48x^{2}+22x}{-48}=-\frac{100}{-48}$$

Dividing by -48 undoes the multiplication by -48.

$$x^{2}+\frac{22}{-48}x=-\frac{100}{-48}$$

Reduce the fraction \frac{22}{-48} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{11}{24}x=-\frac{100}{-48}$$

Reduce the fraction \frac{-100}{-48} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{11}{24}x=\frac{25}{12}$$

Divide -\frac{11}{24}, the coefficient of the x term, by 2 to get -\frac{11}{48}. Then add the square of -\frac{11}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{24}x+\left(-\frac{11}{48}\right)^{2}=\frac{25}{12}+\left(-\frac{11}{48}\right)^{2}$$

Square -\frac{11}{48} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{24}x+\frac{121}{2304}=\frac{25}{12}+\frac{121}{2304}$$

Add \frac{25}{12} to \frac{121}{2304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{24}x+\frac{121}{2304}=\frac{4921}{2304}$$

Factor x^{2}-\frac{11}{24}x+\frac{121}{2304}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{48}\right)^{2}=\frac{4921}{2304}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{48}\right)^{2}}=\sqrt{\frac{4921}{2304}}$$

Simplify.

$$x-\frac{11}{48}=\frac{\sqrt{4921}}{48} x-\frac{11}{48}=-\frac{\sqrt{4921}}{48}$$

Add \frac{11}{48} to both sides of the equation.

$$x=\frac{\sqrt{4921}+11}{48} x=\frac{11-\sqrt{4921}}{48}$$