Solve for m 4m^2-21m+10=-5m^2

$4 m^{2} - 21 m + 10 = - 5 m^{2}$

Step 1

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

Add $5 m^{2}$ to both sides of the equation.

$4 m^{2} - 21 m + 10 + 5 m^{2} = 0$

Add $4 m^{2}$ and $5 m^{2}$ .

$9 m^{2} - 21 m + 10 = 0$

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 $a \cdot c = 9 \cdot 10 = 90$ and whose sum is $b = - 21$ .

Factor $- 21$ out of $- 21 m$ .

$9 m^{2} - 21 m + 10 = 0$

Rewrite $- 21$ as $- 6$ plus $- 15$

$9 m^{2} + (- 6 - 15) m + 10 = 0$

Apply the distributive property.

$9 m^{2} - 6 m - 15 m + 10 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 m^{2} - 6 m) - 15 m + 10 = 0$

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

$3 m (3 m - 2) - 5 (3 m - 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 m - 2$ .

$(3 m - 2) (3 m - 5) = 0$

Step 3

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

$3 m - 2 = 0$

$3 m - 5 = 0$

Step 4

Set $3 m - 2$ equal to $0$ and solve for $m$ .

Set $3 m - 2$ equal to $0$ .

$3 m - 2 = 0$

Solve $3 m - 2 = 0$ for $m$ .

Add $2$ to both sides of the equation.

$3 m = 2$

Divide each term in $3 m = 2$ by $3$ and simplify.

Divide each term in $3 m = 2$ by $3$ .

$\frac{3 m}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 m}{3} = \frac{2}{3}$

Divide $m$ by $1$ .

$m = \frac{2}{3}$

Step 5

Set $3 m - 5$ equal to $0$ and solve for $m$ .

Set $3 m - 5$ equal to $0$ .

$3 m - 5 = 0$

Solve $3 m - 5 = 0$ for $m$ .

Add $5$ to both sides of the equation.

$3 m = 5$

Divide each term in $3 m = 5$ by $3$ and simplify.

Divide each term in $3 m = 5$ by $3$ .

$\frac{3 m}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 m}{3} = \frac{5}{3}$

Divide $m$ by $1$ .

$m = \frac{5}{3}$

Step 6

The final solution is all the values that make $(3 m - 2) (3 m - 5) = 0$ true.

$m = \frac{2}{3}, \frac{5}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$m = \frac{2}{3}, \frac{5}{3}$

Decimal Form:

$m = 0.\overline{6}, 1.\overline{6}$

Mixed Number Form:

$m = \frac{2}{3}, 1 \frac{2}{3}$

Solve for m 4m^2-21m+10=-5m^2

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