Trigonometry Examples

$2 x^{2} + 5 x - 7 = 0$

Step 1

Factor by grouping.

Tap for more steps...

Step 1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 7 = - 14$ and whose sum is $b = 5$ .

Tap for more steps...

Step 1.1.1

Factor $5$ out of $5 x$ .

$2 x^{2} + 5 (x) - 7 = 0$

Step 1.1.2

Rewrite $5$ as $- 2$ plus $7$

$2 x^{2} + (- 2 + 7) x - 7 = 0$

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.1

Group the first two terms and the last two terms.

$(2 x^{2} - 2 x) + 7 x - 7 = 0$

Step 1.2.2

Factor out the greatest common factor (GCF) from each group.

$2 x (x - 1) + 7 (x - 1) = 0$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (2 x + 7) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$2 x + 7 = 0$

Step 3

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

Set $2 x + 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Step 4.2

Solve $2 x + 7 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Step 4.2.2

Divide each term in $2 x = - 7$ by $2$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in $2 x = - 7$ by $2$ .

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

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.2.3.1

Move the negative in front of the fraction.

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

Step 5

The final solution is all the values that make $(x - 1) (2 x + 7) = 0$ true.

$x = 1, - \frac{7}{2}$