Problem 54 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate. $$2 x^{2}-7 x=-5$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5}{2}\) and \(x = 1\).

Step 1: Write the equation in standard form

The given equation is \(2x^2 - 7x = -5\). First, move all terms to one side to write the equation in the standard quadratic form, \(ax^2 + bx + c = 0\). Add 5 to both sides to get: \[2x^2 - 7x + 5 = 0\]

Step 2: Identify coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(2x^2 - 7x + 5 = 0\). Here, \(a = 2\), \(b = -7\), and \(c = 5\).

Step 3: Use the quadratic formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the coefficients \(a = 2\), \(b = -7\), and \(c = 5\) into the formula. Calculate the discriminant \(b^2 - 4ac\) first.

Step 4: Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\): \[(-7)^2 - 4 imes 2 imes 5 = 49 - 40 = 9\]The discriminant is 9.

Step 5: Substitute and solve for x

Use the quadratic formula with the discriminant: \[x = \frac{-(-7) \pm \sqrt{9}}{2 imes 2}\]Simplify:\[x = \frac{7 \pm 3}{4}\]This gives two solutions:\(x = \frac{10}{4} = \frac{5}{2}\) and \(x = \frac{4}{4} = 1\).

Step 6: Verify the solutions

Substitute \(x = \frac{5}{2}\) and \(x = 1\) back into the original equation to ensure they are correct. For \(x = 1\): \[2(1)^2 - 7(1) = 2 - 7 = -5\]For \(x = \frac{5}{2}\): \[2(\frac{5}{2})^2 - 7(\frac{5}{2}) = 2(\frac{25}{4}) - \frac{35}{2} = \frac{50}{4} - \frac{70}{4} = -5\]Both solutions satisfy the original equation.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations, which are in the form of \(ax^2 + bx + c = 0\). This formula allows you to find the values of \(x\) without needing to factor the equation, making it highly versatile. The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. The "\(\pm\)" symbol indicates that there can be two solutions: one using addition and the other using subtraction. This aspect of the formula is crucial because quadratic equations often have two solutions. Understanding this formula will enable you to tackle a wide variety of problems involving quadratic equations.

Discriminant

The discriminant is a key part of solving quadratic equations using the quadratic formula, and it is found inside the square root of the formula: \(b^2 - 4ac\). The discriminant helps determine the nature of the roots of the quadratic equation:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, known as a repeated or double root.
• If \(b^2 - 4ac < 0\), there are no real roots, but two complex conjugate roots.

Understanding the discriminant provides insight into the number and type of solutions, allowing one to predict the behavior of the quadratic before fully solving it.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. Having the equation in this form is essential for using the quadratic formula effectively.
To convert a quadratic equation into standard form, you need to collect all terms on one side of the equation, ensuring the equation equals zero.

• For example, convert \(2x^2 - 7x = -5\) into \(2x^2 - 7x + 5 = 0\).

Once in standard form, you can easily identify the coefficients \(a\), \(b\), and \(c\), which are necessary for applying the quadratic formula. Standardizing the equation is the initial step towards solving it.

Solving Quadratics

To solve a quadratic equation, begin by ensuring it is in the standard form \(ax^2 + bx + c = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.
First, calculate the discriminant \(b^2 - 4ac\). In our problem, this was 9. With this information, substitute back into the quadratic formula. The process involves operating the following steps:

• Compute \(\sqrt{b^2 - 4ac}\).
• Calculate \(-b \pm \sqrt{b^2 - 4ac}\).
• Divide the result by \(2a\) to find the possible values of \(x\).

In this case, we found two solutions: \(x = \frac{5}{2}\) and \(x = 1\). You should always solve for both the positive and negative scenarios in the "\(\pm\)" part to find all possible solutions.

Verification of Solutions

Verification of solutions in quadratic equations confirms the correctness of the calculated roots. After solving the quadratic, substitute each found solution back into the original equation to verify.
For example, consider the solutions \(x = \frac{5}{2}\) and \(x = 1\) for the equation \(2x^2 - 7x + 5 = 0\). Substitute each back into the equation to see if you reach zero:

• Substitute \(x = 1\): \(2(1)^2 - 7(1) + 5 = 0\)
• Substitute \(x = \frac{5}{2}\): \(2(\frac{5}{2})^2 - 7(\frac{5}{2}) + 5 = 0\)

Both substitutions yield zero, verifying the solutions. This confirmation step ensures that there were no mistakes during calculations and instills confidence in the accuracy of the result.