Solve x^2+49=14x

$$x ^ { 2 } + 49 = 14 x$$

Solve for x

$$x=7$$

Steps Using Factoring

Subtract 14x from both sides.

$$x^{2}+49-14x=0$$

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

$$x^{2}-14x+49=0$$

To solve the equation, factor x^{2}-14x+49 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.

$$a+b=-14 ab=49$$

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 49.

$$-1,-49 -7,-7$$

Calculate the sum for each pair.

$$-1-49=-50 -7-7=-14$$

The solution is the pair that gives sum -14.

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

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

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

Rewrite as a binomial square.

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

To find equation solution, solve x-7=0.

$$x=7$$

Steps Using Factoring By Grouping

Subtract 14x from both sides.

$$x^{2}+49-14x=0$$

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

$$x^{2}-14x+49=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+49. To find a and b, set up a system to be solved.

$$a+b=-14 ab=1\times 49=49$$

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 49.

$$-1,-49 -7,-7$$

Calculate the sum for each pair.

$$-1-49=-50 -7-7=-14$$

The solution is the pair that gives sum -14.

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

Rewrite x^{2}-14x+49 as \left(x^{2}-7x\right)+\left(-7x+49\right).

$$\left(x^{2}-7x\right)+\left(-7x+49\right)$$

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

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

Factor out common term x-7 by using distributive property.

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

Rewrite as a binomial square.

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

To find equation solution, solve x-7=0.

$$x=7$$

Steps Using the Quadratic Formula

Subtract 14x from both sides.

$$x^{2}+49-14x=0$$

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.

$$x^{2}-14x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 49}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 49}}{2}$$

Multiply -4 times 49.

$$x=\frac{-\left(-14\right)±\sqrt{196-196}}{2}$$

Add 196 to -196.

$$x=\frac{-\left(-14\right)±\sqrt{0}}{2}$$

Take the square root of 0.

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

The opposite of -14 is 14.

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

Divide 14 by 2.

$$x=7$$

Steps for Completing the Square

Subtract 14x from both sides.

$$x^{2}+49-14x=0$$

Subtract 49 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-14x=-49$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-49+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-49+49$$

Add -49 to 49.

$$x^{2}-14x+49=0$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-7=0 x-7=0$$

Add 7 to both sides of the equation.

$$x=7 x=7$$

The equation is now solved. Solutions are the same.

$$x=7$$