Solve 34x+23x^2-50x=0

$$34 x + 23 x ^ { 2 } - 50 x = 0$$

Solve for x

$$x=\frac{16}{23}\approx 0.695652174$$

$$x=0$$

Steps Using Factoring

Factor out x.

$$x\left(34+23x-50\right)=0$$

To find equation solutions, solve x=0 and -16+23x=0.

$$x=0 x=\frac{16}{23}$$

Steps Using the Quadratic Formula

Combine 34x and -50x to get -16x.

$$-16x+23x^{2}=0$$

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.

$$23x^{2}-16x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 23 for a, -16 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}}}{2\times 23}$$

Take the square root of \left(-16\right)^{2}.

$$x=\frac{-\left(-16\right)±16}{2\times 23}$$

The opposite of -16 is 16.

$$x=\frac{16±16}{2\times 23}$$

Multiply 2 times 23.

$$x=\frac{16±16}{46}$$

Now solve the equation x=\frac{16±16}{46} when ± is plus. Add 16 to 16.

$$x=\frac{32}{46}$$

Reduce the fraction \frac{32}{46} to lowest terms by extracting and canceling out 2.

$$x=\frac{16}{23}$$

Now solve the equation x=\frac{16±16}{46} when ± is minus. Subtract 16 from 16.

$$x=\frac{0}{46}$$

Divide 0 by 46.

$$x=0$$

The equation is now solved.

$$x=\frac{16}{23} x=0$$

Steps for Completing the Square

Combine 34x and -50x to get -16x.

$$-16x+23x^{2}=0$$

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.

$$23x^{2}-16x=0$$

Divide both sides by 23.

$$\frac{23x^{2}-16x}{23}=\frac{0}{23}$$

Dividing by 23 undoes the multiplication by 23.

$$x^{2}-\frac{16}{23}x=\frac{0}{23}$$

Divide 0 by 23.

$$x^{2}-\frac{16}{23}x=0$$

Divide -\frac{16}{23}, the coefficient of the x term, by 2 to get -\frac{8}{23}. Then add the square of -\frac{8}{23} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{23}x+\left(-\frac{8}{23}\right)^{2}=\left(-\frac{8}{23}\right)^{2}$$

Square -\frac{8}{23} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{23}x+\frac{64}{529}=\frac{64}{529}$$

Factor x^{2}-\frac{16}{23}x+\frac{64}{529}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{23}\right)^{2}=\frac{64}{529}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{23}\right)^{2}}=\sqrt{\frac{64}{529}}$$

Simplify.

$$x-\frac{8}{23}=\frac{8}{23} x-\frac{8}{23}=-\frac{8}{23}$$

Add \frac{8}{23} to both sides of the equation.

$$x=\frac{16}{23} x=0$$