Solve x^2-10x+35=7x-35

$$x ^ { 2 } - 10 x + 35 = 7 x - 35$$

Solve for x

$$x=7$$

$$x=10$$

Steps Using Factoring

Subtract 7x from both sides.

$$x^{2}-10x+35-7x=-35$$

Combine -10x and -7x to get -17x.

$$x^{2}-17x+35=-35$$

Add 35 to both sides.

$$x^{2}-17x+35+35=0$$

Add 35 and 35 to get 70.

$$x^{2}-17x+70=0$$

To solve the equation, factor x^{2}-17x+70 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-17 ab=70$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 70.

$$-1,-70 -2,-35 -5,-14 -7,-10$$

Calculate the sum for each pair.

$$-1-70=-71 -2-35=-37 -5-14=-19 -7-10=-17$$

The solution is the pair that gives sum -17.

$$a=-10 b=-7$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-10\right)\left(x-7\right)$$

To find equation solutions, solve x-10=0 and x-7=0.

$$x=10 x=7$$

Steps Using Factoring By Grouping

Subtract 7x from both sides.

$$x^{2}-10x+35-7x=-35$$

Combine -10x and -7x to get -17x.

$$x^{2}-17x+35=-35$$

Add 35 to both sides.

$$x^{2}-17x+35+35=0$$

Add 35 and 35 to get 70.

$$x^{2}-17x+70=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+70. To find a and b, set up a system to be solved.

$$a+b=-17 ab=1\times 70=70$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 70.

$$-1,-70 -2,-35 -5,-14 -7,-10$$

Calculate the sum for each pair.

$$-1-70=-71 -2-35=-37 -5-14=-19 -7-10=-17$$

The solution is the pair that gives sum -17.

$$a=-10 b=-7$$

Rewrite x^{2}-17x+70 as \left(x^{2}-10x\right)+\left(-7x+70\right).

$$\left(x^{2}-10x\right)+\left(-7x+70\right)$$

Factor out x in the first and -7 in the second group.

$$x\left(x-10\right)-7\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x-7\right)$$

To find equation solutions, solve x-10=0 and x-7=0.

$$x=10 x=7$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

$$x^{2}-10x+35-7x=-35$$

Combine -10x and -7x to get -17x.

$$x^{2}-17x+35=-35$$

Add 35 to both sides.

$$x^{2}-17x+35+35=0$$

Add 35 and 35 to get 70.

$$x^{2}-17x+70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -17 for b, and 70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 70}}{2}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 70}}{2}$$

Multiply -4 times 70.

$$x=\frac{-\left(-17\right)±\sqrt{289-280}}{2}$$

Add 289 to -280.

$$x=\frac{-\left(-17\right)±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-\left(-17\right)±3}{2}$$

The opposite of -17 is 17.

$$x=\frac{17±3}{2}$$

Now solve the equation x=\frac{17±3}{2} when ± is plus. Add 17 to 3.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

Now solve the equation x=\frac{17±3}{2} when ± is minus. Subtract 3 from 17.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

The equation is now solved.

$$x=10 x=7$$

Steps for Completing the Square

Subtract 7x from both sides.

$$x^{2}-10x+35-7x=-35$$

Combine -10x and -7x to get -17x.

$$x^{2}-17x+35=-35$$

Subtract 35 from both sides.

$$x^{2}-17x=-35-35$$

Subtract 35 from -35 to get -70.

$$x^{2}-17x=-70$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-70+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=-70+\frac{289}{4}$$

Add -70 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{9}{4}$$

Factor x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{3}{2} x-\frac{17}{2}=-\frac{3}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=10 x=7$$