Solve 3x(4-x)=10x^2-9x

$$3 x ( 4 - x ) = 10 x ^ { 2 } - 9 x$$

Solve for x

$$x = \frac{21}{13} = 1\frac{8}{13} \approx 1.615384615$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 3x by 4-x.

$$12x-3x^{2}=10x^{2}-9x$$

Subtract 10x^{2} from both sides.

$$12x-3x^{2}-10x^{2}=-9x$$

Combine -3x^{2} and -10x^{2} to get -13x^{2}.

$$12x-13x^{2}=-9x$$

Add 9x to both sides.

$$12x-13x^{2}+9x=0$$

Combine 12x and 9x to get 21x.

$$21x-13x^{2}=0$$

Factor out x.

$$x\left(21-13x\right)=0$$

To find equation solutions, solve x=0 and 21-13x=0.

$$x=0 x=\frac{21}{13}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by 4-x.

$$12x-3x^{2}=10x^{2}-9x$$

Subtract 10x^{2} from both sides.

$$12x-3x^{2}-10x^{2}=-9x$$

Combine -3x^{2} and -10x^{2} to get -13x^{2}.

$$12x-13x^{2}=-9x$$

Add 9x to both sides.

$$12x-13x^{2}+9x=0$$

Combine 12x and 9x to get 21x.

$$21x-13x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-13x^{2}+21x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13 for a, 21 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}}}{2\left(-13\right)}$$

Take the square root of 21^{2}.

$$x=\frac{-21±21}{2\left(-13\right)}$$

Multiply 2 times -13.

$$x=\frac{-21±21}{-26}$$

Now solve the equation x=\frac{-21±21}{-26} when ± is plus. Add -21 to 21.

$$x=\frac{0}{-26}$$

Divide 0 by -26.

$$x=0$$

Now solve the equation x=\frac{-21±21}{-26} when ± is minus. Subtract 21 from -21.

$$x=-\frac{42}{-26}$$

Reduce the fraction \frac{-42}{-26} to lowest terms by extracting and canceling out 2.

$$x=\frac{21}{13}$$

The equation is now solved.

$$x=0 x=\frac{21}{13}$$

Steps for Completing the Square

Use the distributive property to multiply 3x by 4-x.

$$12x-3x^{2}=10x^{2}-9x$$

Subtract 10x^{2} from both sides.

$$12x-3x^{2}-10x^{2}=-9x$$

Combine -3x^{2} and -10x^{2} to get -13x^{2}.

$$12x-13x^{2}=-9x$$

Add 9x to both sides.

$$12x-13x^{2}+9x=0$$

Combine 12x and 9x to get 21x.

$$21x-13x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-13x^{2}+21x=0$$

Divide both sides by -13.

$$\frac{-13x^{2}+21x}{-13}=\frac{0}{-13}$$

Dividing by -13 undoes the multiplication by -13.

$$x^{2}+\frac{21}{-13}x=\frac{0}{-13}$$

Divide 21 by -13.

$$x^{2}-\frac{21}{13}x=\frac{0}{-13}$$

Divide 0 by -13.

$$x^{2}-\frac{21}{13}x=0$$

Divide -\frac{21}{13}, the coefficient of the x term, by 2 to get -\frac{21}{26}. Then add the square of -\frac{21}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{21}{13}x+\left(-\frac{21}{26}\right)^{2}=\left(-\frac{21}{26}\right)^{2}$$

Square -\frac{21}{26} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{21}{13}x+\frac{441}{676}=\frac{441}{676}$$

Factor x^{2}-\frac{21}{13}x+\frac{441}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{26}\right)^{2}=\frac{441}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{26}\right)^{2}}=\sqrt{\frac{441}{676}}$$

Simplify.

$$x-\frac{21}{26}=\frac{21}{26} x-\frac{21}{26}=-\frac{21}{26}$$

Add \frac{21}{26} to both sides of the equation.

$$x=\frac{21}{13} x=0$$