Problem 39 $$ \text { Solve the given quadr... [FREE SOLUTION]

Chapter 7: Problem 39

$$ \text { Solve the given quadratic equations by factoring.}$$ In Eq. $(7.1),$ for $a=2, b=-7,$ and $c=3,$ show that the sum of the roots is $-b / a$

Short Answer

Expert verified

The sum of the roots $\frac{7}{2}$ is indeed equal to $-b/a$.

Step by step solution

Write the quadratic equation

Given the quadratic equation in the standard form \[ ax^2 + bx + c = 0 \]Substitute the provided values into the equation:\[ 2x^2 - 7x + 3 = 0 \]

Factor the quadratic equation

To factor the equation, look for two numbers that multiply to $a \cdot c = 2 \cdot 3 = 6$ and add to $b = -7$. These numbers are $-1$ and $-6$. Thus, rewrite the equation:\[ 2x^2 - x - 6x + 3 = 0 \] Next, group terms and factor them:\[ (2x^2 - x) + (-6x + 3) = 0 \] \[ x(2x - 1) - 3(2x - 1) = 0 \] Now factor by grouping:\[ (x - 3)(2x - 1) = 0 \]

Identify the roots

Set each factor equal to zero and solve for $x$:\[ x - 3 = 0 \quad \text{or} \quad 2x - 1 = 0 \]Solving these:\[ x = 3 \quad \text{and} \quad x = \frac{1}{2} \]

Sum the roots

Now add the roots from Step 3:\[ 3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2} \]

Calculate $(-b)/a$

Substitute the given values into the formula $(-b)/a$ to check the sum of roots.\[ \text{Sum of roots} = -\frac{-7}{2} = \frac{7}{2} \]

Verification

The calculated sum of the roots ($\frac{7}{2}$) matches the value $-b/a$ ($\frac{7}{2}$), confirming the relationship.

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Key Concepts

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

Factoring

Factoring is a crucial method for solving quadratic equations. The goal is to express the equation as a product of two binomials. For our equation, we are dealing with the quadratic form:

$ ax^2 + bx + c = 0 $

To factor it, we seek two numbers that multiply to $ a \cdot c $ and add up to $ b $. These numbers help break the middle term to simplify factoring. For example, in our quadratic equation $ 2x^2 - 7x + 3 = 0 $, we identified $-1$ and $-6$ because:

They multiply to $ 6 $ (since $ 2 \cdot 3 = 6 $)
They add to $-7$ (which is the given $ b $)

Rewriting the middle term in terms of these numbers allows the quadratic to be factored effectively into $ (x - 3)(2x - 1) = 0 $. Once factored, solving becomes simple by using the Zero Product Property.

Sum of Roots

The sum of roots of a quadratic equation offers insights into the relationship between 'b' and 'a'. In a standard quadratic equation:

$ ax^2 + bx + c = 0 $

the sum of its roots can be found using the formula $ -b/a $. It arises from Vieta's formulas, which link coefficients of polynomials to sums and products of their roots.
For our example, substituting in our given values $ a = 2 $ and $ b = -7 $ results in:

Sum of roots = $ -(-7)/2 = 7/2 $

Here's a deeper look:

Negative signs might appear twice, once from the formula and once from the sign of $ b $.
The outcome ensures it reflects the actual sum of roots derived from the factors.

Standard Form

The standard form of a quadratic equation is fundamental for factoring and finding roots. It follows the structure:

$ ax^2 + bx + c = 0 $

Making sure an equation is in this form is the first step in solving it.
In the given problem, substituting the values $ a = 2 $, $ b = -7 $, and $ c = 3 $ results in the equation $ 2x^2 - 7x + 3 = 0 $. This step is critical:

It ensures that all coefficients are correctly placed before the factoring process.
It validates whether a solution pertains to real roots by indicating whether a solution is possible or if complex numbers may come into play.

Being in standard form prepares the equation for both solving and plotting on a graph if needed.

Roots

The roots of a quadratic equation are its solutions. They are the values of $ x $ that satisfy the equation when plugged back in. For instance, once factored, our equation $ (x - 3)(2x - 1) = 0 $ is straightforward to solve.
To find the roots, apply the Zero Product Property:

If $ (a)(b) = 0 $, then $ a = 0 $ or $ b = 0 $.

Translate this to our factors:

$ x - 3 = 0 \Rightarrow x = 3 $
$ 2x - 1 = 0 \Rightarrow x = \frac{1}{2} $

These values $ x = 3 $ and $ x = \frac{1}{2} $ are the roots. Verifying roots involves substituting back into the original equation to ensure they provide true statements. Roots are where the equation's graph intersects the x-axis, further aiding in visual representations of solutions.

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91影视

$$ \text { Solve the given quadratic equations by factoring.}$$ In Eq. \((7.1),\) for \(a=2, b=-7,\) and \(c=3,\) show that the sum of the roots is \(-b / a\)

Short Answer

Step by step solution

Write the quadratic equation

Factor the quadratic equation

Identify the roots

Sum the roots

Calculate \((-b)/a\)

Verification

Key Concepts

Factoring

Sum of Roots

Standard Form

Roots

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