Solve x(15x-23)-4(15x-23)=0

$$x(15x-23)-4(15x-23)=0$$

Solve for x

$$x=4$$

$$x = \frac{23}{15} = 1\frac{8}{15} \approx 1.533333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 15x-23.

$$15x^{2}-23x-4\left(15x-23\right)=0$$

Use the distributive property to multiply -4 by 15x-23.

$$15x^{2}-23x-60x+92=0$$

Combine -23x and -60x to get -83x.

$$15x^{2}-83x+92=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -83 for b, and 92 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-83\right)±\sqrt{\left(-83\right)^{2}-4\times 15\times 92}}{2\times 15}$$

Square -83.

$$x=\frac{-\left(-83\right)±\sqrt{6889-4\times 15\times 92}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-83\right)±\sqrt{6889-60\times 92}}{2\times 15}$$

Multiply -60 times 92.

$$x=\frac{-\left(-83\right)±\sqrt{6889-5520}}{2\times 15}$$

Add 6889 to -5520.

$$x=\frac{-\left(-83\right)±\sqrt{1369}}{2\times 15}$$

Take the square root of 1369.

$$x=\frac{-\left(-83\right)±37}{2\times 15}$$

The opposite of -83 is 83.

$$x=\frac{83±37}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{83±37}{30}$$

Now solve the equation x=\frac{83±37}{30} when ± is plus. Add 83 to 37.

$$x=\frac{120}{30}$$

Divide 120 by 30.

$$x=4$$

Now solve the equation x=\frac{83±37}{30} when ± is minus. Subtract 37 from 83.

$$x=\frac{46}{30}$$

Reduce the fraction \frac{46}{30} to lowest terms by extracting and canceling out 2.

$$x=\frac{23}{15}$$

The equation is now solved.

$$x=4 x=\frac{23}{15}$$

Steps for Completing the Square

Use the distributive property to multiply x by 15x-23.

$$15x^{2}-23x-4\left(15x-23\right)=0$$

Use the distributive property to multiply -4 by 15x-23.

$$15x^{2}-23x-60x+92=0$$

Combine -23x and -60x to get -83x.

$$15x^{2}-83x+92=0$$

Subtract 92 from both sides. Anything subtracted from zero gives its negation.

$$15x^{2}-83x=-92$$

Divide both sides by 15.

$$\frac{15x^{2}-83x}{15}=-\frac{92}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}-\frac{83}{15}x=-\frac{92}{15}$$

Divide -\frac{83}{15}, the coefficient of the x term, by 2 to get -\frac{83}{30}. Then add the square of -\frac{83}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{83}{15}x+\left(-\frac{83}{30}\right)^{2}=-\frac{92}{15}+\left(-\frac{83}{30}\right)^{2}$$

Square -\frac{83}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{83}{15}x+\frac{6889}{900}=-\frac{92}{15}+\frac{6889}{900}$$

Add -\frac{92}{15} to \frac{6889}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{83}{15}x+\frac{6889}{900}=\frac{1369}{900}$$

Factor x^{2}-\frac{83}{15}x+\frac{6889}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{83}{30}\right)^{2}=\frac{1369}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{83}{30}\right)^{2}}=\sqrt{\frac{1369}{900}}$$

Simplify.

$$x-\frac{83}{30}=\frac{37}{30} x-\frac{83}{30}=-\frac{37}{30}$$

Add \frac{83}{30} to both sides of the equation.

$$x=4 x=\frac{23}{15}$$