Problem 39 - Basic Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Write the quadratic equation

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

Step 2: 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:\[ (x - 3)(2x - 1) = 0 \]

Step 3: 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:\[ x = 3 \quad \text{and} \quad x = \frac{1}{2} \]

Step 4: Sum the roots

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

Step 5: Calculate \((-b)/a\)

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

Step 6: Verification

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

Key Concepts

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.