Identify the solution(s) of m² + 10 = 15m - 34.
Understand the Problem
The question asks to identify the solution(s) of the quadratic equation m² + 10 = 15m - 34. To solve this, we need to rearrange the equation into the standard quadratic form (ax² + bx + c = 0) and then find the roots (solutions) of the equation, which are the values of 'm' that satisfy the equation. We can use factoring, completing the square, or the quadratic formula to find the solutions.
Answer
$m = 4, 11$
Answer for screen readers
The solutions are $m = 4$ and $m = 11$.
Steps to Solve
- Rearrange the equation
We want to get the equation into the standard quadratic form, which is $ax^2 + bx + c = 0$. In our case, the variable is $m$. So, we need to rearrange $m^2 + 10 = 15m - 34$ to look like $am^2 + bm + c = 0$.
Subtract $15m$ from both sides:
$m^2 - 15m + 10 = -34$
Add $34$ to both sides:
$m^2 - 15m + 44 = 0$
- Factor the quadratic equation
Now, we need to factor the quadratic equation $m^2 - 15m + 44 = 0$. We are looking for two numbers that multiply to $44$ and add up to $-15$. These numbers are $-4$ and $-11$ since $(-4) \times (-11) = 44$ and $(-4) + (-11) = -15$.
Therefore, we can factor the equation as follows:
$(m - 4)(m - 11) = 0$
- Solve for m
To find the solutions, we set each factor equal to zero:
$m - 4 = 0$ or $m - 11 = 0$
Solving for $m$ in each case gives:
$m = 4$ or $m = 11$
The solutions are $m = 4$ and $m = 11$.
More Information
Quadratic equations can have two, one, or zero real solutions. In this case, we found two distinct real solutions.
Tips
A common mistake is making sign errors when rearranging the equation into standard form.
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