Solve 126=(x-10)(35-x)

$$126=(x-10)(35-x)$$

Solve for x

$$x=17$$

$$x=28$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-10 by 35-x and combine like terms.

$$126=45x-x^{2}-350$$

Swap sides so that all variable terms are on the left hand side.

$$45x-x^{2}-350=126$$

Subtract 126 from both sides.

$$45x-x^{2}-350-126=0$$

Subtract 126 from -350 to get -476.

$$45x-x^{2}-476=0$$

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.

$$-x^{2}+45x-476=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 45 for b, and -476 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-1\right)\left(-476\right)}}{2\left(-1\right)}$$

Square 45.

$$x=\frac{-45±\sqrt{2025-4\left(-1\right)\left(-476\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-45±\sqrt{2025+4\left(-476\right)}}{2\left(-1\right)}$$

Multiply 4 times -476.

$$x=\frac{-45±\sqrt{2025-1904}}{2\left(-1\right)}$$

Add 2025 to -1904.

$$x=\frac{-45±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-45±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-45±11}{-2}$$

Now solve the equation x=\frac{-45±11}{-2} when ± is plus. Add -45 to 11.

$$x=-\frac{34}{-2}$$

Divide -34 by -2.

$$x=17$$

Now solve the equation x=\frac{-45±11}{-2} when ± is minus. Subtract 11 from -45.

$$x=-\frac{56}{-2}$$

Divide -56 by -2.

$$x=28$$

The equation is now solved.

$$x=17 x=28$$

Steps for Completing the Square

Use the distributive property to multiply x-10 by 35-x and combine like terms.

$$126=45x-x^{2}-350$$

Swap sides so that all variable terms are on the left hand side.

$$45x-x^{2}-350=126$$

Add 350 to both sides.

$$45x-x^{2}=126+350$$

Add 126 and 350 to get 476.

$$45x-x^{2}=476$$

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.

$$-x^{2}+45x=476$$

Divide both sides by -1.

$$\frac{-x^{2}+45x}{-1}=\frac{476}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{45}{-1}x=\frac{476}{-1}$$

Divide 45 by -1.

$$x^{2}-45x=\frac{476}{-1}$$

Divide 476 by -1.

$$x^{2}-45x=-476$$

Divide -45, the coefficient of the x term, by 2 to get -\frac{45}{2}. Then add the square of -\frac{45}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-45x+\left(-\frac{45}{2}\right)^{2}=-476+\left(-\frac{45}{2}\right)^{2}$$

Square -\frac{45}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-45x+\frac{2025}{4}=-476+\frac{2025}{4}$$

Add -476 to \frac{2025}{4}.

$$x^{2}-45x+\frac{2025}{4}=\frac{121}{4}$$

Factor x^{2}-45x+\frac{2025}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{45}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{45}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{45}{2}=\frac{11}{2} x-\frac{45}{2}=-\frac{11}{2}$$

Add \frac{45}{2} to both sides of the equation.

$$x=28 x=17$$