Solve 1042x-262x^2=232+10x

$$1042 x - 262 x ^ { 2 } = 232 + 10 x$$

Solve for x

$$x = \frac{2 \sqrt{12842} + 258}{131} \approx 3.699580893$$

$$x=\frac{258-2\sqrt{12842}}{131}\approx 0.239350404$$

Steps Using the Quadratic Formula

Subtract 232 from both sides.

$$1042x-262x^{2}-232=10x$$

Subtract 10x from both sides.

$$1042x-262x^{2}-232-10x=0$$

Combine 1042x and -10x to get 1032x.

$$1032x-262x^{2}-232=0$$

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.

$$-262x^{2}+1032x-232=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -262 for a, 1032 for b, and -232 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1032±\sqrt{1032^{2}-4\left(-262\right)\left(-232\right)}}{2\left(-262\right)}$$

Square 1032.

$$x=\frac{-1032±\sqrt{1065024-4\left(-262\right)\left(-232\right)}}{2\left(-262\right)}$$

Multiply -4 times -262.

$$x=\frac{-1032±\sqrt{1065024+1048\left(-232\right)}}{2\left(-262\right)}$$

Multiply 1048 times -232.

$$x=\frac{-1032±\sqrt{1065024-243136}}{2\left(-262\right)}$$

Add 1065024 to -243136.

$$x=\frac{-1032±\sqrt{821888}}{2\left(-262\right)}$$

Take the square root of 821888.

$$x=\frac{-1032±8\sqrt{12842}}{2\left(-262\right)}$$

Multiply 2 times -262.

$$x=\frac{-1032±8\sqrt{12842}}{-524}$$

Now solve the equation x=\frac{-1032±8\sqrt{12842}}{-524} when ± is plus. Add -1032 to 8\sqrt{12842}.

$$x=\frac{8\sqrt{12842}-1032}{-524}$$

Divide -1032+8\sqrt{12842} by -524.

$$x=\frac{258-2\sqrt{12842}}{131}$$

Now solve the equation x=\frac{-1032±8\sqrt{12842}}{-524} when ± is minus. Subtract 8\sqrt{12842} from -1032.

$$x=\frac{-8\sqrt{12842}-1032}{-524}$$

Divide -1032-8\sqrt{12842} by -524.

$$x=\frac{2\sqrt{12842}+258}{131}$$

The equation is now solved.

$$x=\frac{258-2\sqrt{12842}}{131} x=\frac{2\sqrt{12842}+258}{131}$$

Steps for Completing the Square

Subtract 10x from both sides.

$$1042x-262x^{2}-10x=232$$

Combine 1042x and -10x to get 1032x.

$$1032x-262x^{2}=232$$

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.

$$-262x^{2}+1032x=232$$

Divide both sides by -262.

$$\frac{-262x^{2}+1032x}{-262}=\frac{232}{-262}$$

Dividing by -262 undoes the multiplication by -262.

$$x^{2}+\frac{1032}{-262}x=\frac{232}{-262}$$

Reduce the fraction \frac{1032}{-262} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{516}{131}x=\frac{232}{-262}$$

Reduce the fraction \frac{232}{-262} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{516}{131}x=-\frac{116}{131}$$

Divide -\frac{516}{131}, the coefficient of the x term, by 2 to get -\frac{258}{131}. Then add the square of -\frac{258}{131} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{516}{131}x+\left(-\frac{258}{131}\right)^{2}=-\frac{116}{131}+\left(-\frac{258}{131}\right)^{2}$$

Square -\frac{258}{131} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{516}{131}x+\frac{66564}{17161}=-\frac{116}{131}+\frac{66564}{17161}$$

Add -\frac{116}{131} to \frac{66564}{17161} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{516}{131}x+\frac{66564}{17161}=\frac{51368}{17161}$$

Factor x^{2}-\frac{516}{131}x+\frac{66564}{17161}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{258}{131}\right)^{2}=\frac{51368}{17161}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{258}{131}\right)^{2}}=\sqrt{\frac{51368}{17161}}$$

Simplify.

$$x-\frac{258}{131}=\frac{2\sqrt{12842}}{131} x-\frac{258}{131}=-\frac{2\sqrt{12842}}{131}$$

Add \frac{258}{131} to both sides of the equation.

$$x=\frac{2\sqrt{12842}+258}{131} x=\frac{258-2\sqrt{12842}}{131}$$