Solve 7x^2-42x-49=0

$$7 { x }^{ 2 } -42x-49=0$$

Solve for x

$$x=-1$$

$$x=7$$

Steps Using Factoring By Grouping

Divide both sides by 7.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-7\right)=-7$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-7 b=1$$

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

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

Factor out x in x^{2}-7x.

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

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

$$\left(x-7\right)\left(x+1\right)$$

To find equation solutions, solve x-7=0 and x+1=0.

$$x=7 x=-1$$

Steps Using the Quadratic Formula

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.

$$7x^{2}-42x-49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -42 for b, and -49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 7\left(-49\right)}}{2\times 7}$$

Square -42.

$$x=\frac{-\left(-42\right)±\sqrt{1764-4\times 7\left(-49\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-42\right)±\sqrt{1764-28\left(-49\right)}}{2\times 7}$$

Multiply -28 times -49.

$$x=\frac{-\left(-42\right)±\sqrt{1764+1372}}{2\times 7}$$

Add 1764 to 1372.

$$x=\frac{-\left(-42\right)±\sqrt{3136}}{2\times 7}$$

Take the square root of 3136.

$$x=\frac{-\left(-42\right)±56}{2\times 7}$$

The opposite of -42 is 42.

$$x=\frac{42±56}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{42±56}{14}$$

Now solve the equation x=\frac{42±56}{14} when ± is plus. Add 42 to 56.

$$x=\frac{98}{14}$$

Divide 98 by 14.

$$x=7$$

Now solve the equation x=\frac{42±56}{14} when ± is minus. Subtract 56 from 42.

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

Divide -14 by 14.

$$x=-1$$

The equation is now solved.

$$x=7 x=-1$$

Steps for Completing the Square

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.

$$7x^{2}-42x-49=0$$

Add 49 to both sides of the equation.

$$7x^{2}-42x-49-\left(-49\right)=-\left(-49\right)$$

Subtracting -49 from itself leaves 0.

$$7x^{2}-42x=-\left(-49\right)$$

Subtract -49 from 0.

$$7x^{2}-42x=49$$

Divide both sides by 7.

$$\frac{7x^{2}-42x}{7}=\frac{49}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\left(-\frac{42}{7}\right)x=\frac{49}{7}$$

Divide -42 by 7.

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

Divide 49 by 7.

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

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=7+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=7+9$$

Add 7 to 9.

$$x^{2}-6x+9=16$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-3=4 x-3=-4$$

Add 3 to both sides of the equation.

$$x=7 x=-1$$