Algebra Examples

$x^{2} + 7 x - \frac{1}{2} = 0$

Find the LCD of the terms in the equation.

Tap for more steps...

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 1, 2, 1$

Since $1, 1, 2, 1$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 2, 1$ then find LCM for the variable part .

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 2, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Multiply each term by $2$ and simplify.

Tap for more steps...

Multiply each term in $x^{2} + 7 x - \frac{1}{2} = 0$ by $2$ in order to remove all the denominators from the equation.

$x^{2} \cdot 2 + 7 x \cdot 2 - \frac{1}{2} \cdot 2 = 0 \cdot 2$

Simplify each term.

Tap for more steps...

Move $2$ to the left of $x^{2}$ .

$2 \cdot x^{2} + 7 x \cdot 2 - \frac{1}{2} \cdot 2 = 0 \cdot 2$

Multiply $2$ by $7$ .

$2 x^{2} + 14 x - \frac{1}{2} \cdot 2 = 0 \cdot 2$

Cancel the common factor of $2$ .

Tap for more steps...

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2 x^{2} + 14 x + \frac{- 1}{2} \cdot 2 = 0 \cdot 2$

Write $2$ as a fraction with denominator $1$ .

$2 x^{2} + 14 x + \frac{- 1}{2} \cdot \frac{2}{1} = 0 \cdot 2$

Factor out the greatest common factor $2$ .

$2 x^{2} + 14 x + \frac{- 1}{2 \cdot 1} \cdot \frac{2 \cdot 1}{1} = 0 \cdot 2$

Cancel the common factor.

$2 x^{2} + 14 x + \frac{- 1}{2 \cdot 1} \cdot \frac{2 \cdot 1}{1} = 0 \cdot 2$

Rewrite the expression.

$2 x^{2} + 14 x + \frac{- 1}{1} \cdot \frac{1}{1} = 0 \cdot 2$

Simplify.

Tap for more steps...

Multiply $\frac{- 1}{1}$ and $\frac{1}{1}$ .

$2 x^{2} + 14 x + \frac{- 1}{1} = 0 \cdot 2$

Divide $- 1$ by $1$ .

$2 x^{2} + 14 x - 1 = 0 \cdot 2$

Multiply $0$ by $2$ .

$2 x^{2} + 14 x - 1 = 0$

Solve the equation.

Tap for more steps...

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 2$ , $b = 14$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Simplify.

Tap for more steps...

Simplify the numerator.

Tap for more steps...

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Multiply $2$ by $- 1$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot - 2}}{2 \cdot 2}$

Multiply $- 4$ by $- 2$ .

$x = \frac{- 14 \pm \sqrt{196 + 8}}{2 \cdot 2}$

Add $196$ and $8$ .

$x = \frac{- 14 \pm \sqrt{204}}{2 \cdot 2}$

Rewrite $204$ as $2^{2} \cdot 51$ .

Tap for more steps...

Factor $4$ out of $204$ .

$x = \frac{- 14 \pm \sqrt{4 (51)}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 51}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{51}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 14 \pm 2 \sqrt{51}}{4}$

Simplify $\frac{- 14 \pm 2 \sqrt{51}}{4}$ .

$x = \frac{- 7 \pm \sqrt{51}}{2}$

Factor $- 1$ out of $- 7 \pm \sqrt{51}$ .

$x = \frac{- 17 \pm \sqrt{51}}{2}$

Multiply $- 1$ by $- 1$ .

$x = \frac{1 - 7 \pm \sqrt{51}}{2}$

Multiply $- 7 \pm \sqrt{51}$ by $1$ .

$x = \frac{- 7 \pm \sqrt{51}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{51}}{2}, - \frac{7 + \sqrt{51}}{2}$

$x = \frac{- 7 \pm \sqrt{51}}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 7 \pm \sqrt{51}}{2}$

Decimal Form:

$x = 0.07071421 \dots, - 7.07071421 \dots$

Enter YOUR Problem