Problem 329 Solve. \(4 m^{2}=17 m-15\)... [FREE SOLUTION]

Chapter 7: Problem 329

Solve. \(4 m^{2}=17 m-15\)

Short Answer

Expert verified

\(m = \frac{3}{4}\) or \(m = 5\)

Step by step solution

- Rewrite the Equation

First, rewrite the given quadratic equation in standard form by moving all terms to one side. The standard form is \(ax^2 + bx + c = 0\).

- Form the Equation

Subtract \(17m\) and add \(15\) to both sides of the equation: \[4m^2 - 17m + 15 = 0\]

- Factor the Quadratic Equation

Look for two numbers that multiply to \(4 \times 15 = 60\) and add to \(-17\). After finding those numbers (-12 and -5), factor the quadratic equation: \[4m^2 - 17m + 15 = (4m - 3)(m - 5)\]

- Solve for m

Set each factor equal to zero and solve for \(m\): \[4m - 3 = 0\] \[m = \frac{3}{4}\] \[m - 5 = 0\] \[m = 5\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Quadratic Equations

Factoring is a common method to solve quadratic equations. When we factor a quadratic equation, we are looking for two binomials that, when multiplied together, give the original quadratic expression. This method involves finding two numbers that multiply to give the constant term (in this case, 15) and add to give the coefficient of the middle term (in this case, -17).

Let's take the equation used in the example:

4m^2 - 17m + 15 = 0

In this context, you need to look for two numbers whose product is

4 * 15 = 60

and whose sum is -17. These numbers turn out to be -12 and -5, so the quadratic equation factors into:

(4m - 3)(m - 5) = 0

After factoring, the next step is to solve each binomial equation for m.

Quadratic Equation Standard Form

Quadratic equations should be written in the standard form before attempting to solve them. The standard form of a quadratic equation is:

ax^2 + bx + c = 0

The original equation in the exercise is:

4m^2 = 17m - 15

First, we move all terms to one side to fit the standard form:

4m^2 - 17m + 15 = 0

This format is essential because it makes the equation easier to handle when using different methods to solve it, like factoring, completing the square, or using the quadratic formula. Each method relies on the quadratic being in this form.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of the variable that satisfy the equation. In other words, they're the values of m that make the equation

4m^2 - 17m + 15 = 0

true.

After factoring the quadratic equation, we have:

(4m - 3)(m - 5) = 0

Set each factor equal to zero to find the roots:

4m - 3 = 0

Solving this gives:

m = 3/4

For the other factor:

m - 5 = 0

Solving this gives:

m = 5

So, the roots of the quadratic equation

4m^2 - 17m + 15 = 0

are

m = 3/4

and

m = 5

. These are also known as the solutions of the quadratic equation.

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91影视

Solve. \(4 m^{2}=17 m-15\)

Short Answer

Step by step solution

- Rewrite the Equation

- Form the Equation

- Factor the Quadratic Equation

- Solve for m

Key Concepts

Factoring Quadratic Equations

Quadratic Equation Standard Form

Roots of Quadratic Equations

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