Solve x^2+6=7x

$$x ^ { 2 } + 6 = 7 x$$

Solve for x

$$x=1$$

$$x=6$$

Steps Using Factoring

Subtract 7x from both sides.

$$x^{2}+6-7x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-7x+6=0$$

To solve the equation, factor x^{2}-7x+6 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.

$$a+b=-7 ab=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-6 b=-1$$

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

$$\left(x-6\right)\left(x-1\right)$$

To find equation solutions, solve x-6=0 and x-1=0.

$$x=6 x=1$$

Steps Using Factoring By Grouping

Subtract 7x from both sides.

$$x^{2}+6-7x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}-7x+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=1\times 6=6$$

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 6.

$$-1,-6 -2,-3$$

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-6 b=-1$$

Rewrite x^{2}-7x+6 as \left(x^{2}-6x\right)+\left(-x+6\right).

$$\left(x^{2}-6x\right)+\left(-x+6\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-6\right)-\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x-1\right)$$

To find equation solutions, solve x-6=0 and x-1=0.

$$x=6 x=1$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

$$x^{2}+6-7x=0$$

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.

$$x^{2}-7x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24}}{2}$$

Add 49 to -24.

$$x=\frac{-\left(-7\right)±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-\left(-7\right)±5}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±5}{2}$$

Now solve the equation x=\frac{7±5}{2} when ± is plus. Add 7 to 5.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Now solve the equation x=\frac{7±5}{2} when ± is minus. Subtract 5 from 7.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=6 x=1$$

Steps for Completing the Square

Subtract 7x from both sides.

$$x^{2}+6-7x=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-7x=-6$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-6+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-6+\frac{49}{4}$$

Add -6 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{25}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{5}{2} x-\frac{7}{2}=-\frac{5}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=6 x=1$$