Solve for x 7x^2-14x+5=-x^2

$7 x^{2} - 14 x + 5 = - x^{2}$

Step 1

Move all terms containing $x$ to the left side of the equation.

Add $x^{2}$ to both sides of the equation.

$7 x^{2} - 14 x + 5 + x^{2} = 0$

Add $7 x^{2}$ and $x^{2}$ .

$8 x^{2} - 14 x + 5 = 0$

Step 2

Factor by grouping.

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 $a \cdot c = 8 \cdot 5 = 40$ and whose sum is $b = - 14$ .

Factor $- 14$ out of $- 14 x$ .

$8 x^{2} - 14 x + 5 = 0$

Rewrite $- 14$ as $- 4$ plus $- 10$

$8 x^{2} + (- 4 - 10) x + 5 = 0$

Apply the distributive property.

$8 x^{2} - 4 x - 10 x + 5 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(8 x^{2} - 4 x) - 10 x + 5 = 0$

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

$4 x (2 x - 1) - 5 (2 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(2 x - 1) (4 x - 5) = 0$

Step 3

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

$2 x - 1 = 0$

$4 x - 5 = 0$

Step 4

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 5

Set $4 x - 5$ equal to $0$ and solve for $x$ .

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Solve $4 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$4 x = 5$

Divide each term in $4 x = 5$ by $4$ and simplify.

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 6

The final solution is all the values that make $(2 x - 1) (4 x - 5) = 0$ true.

$x = \frac{1}{2}, \frac{5}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, \frac{5}{4}$

Decimal Form:

$x = 0.5, 1.25$

Mixed Number Form:

$x = \frac{1}{2}, 1 \frac{1}{4}$

Solve for x 7x^2-14x+5=-x^2

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