Problem 74 - College Algebra with Modeling and Visualization

Question

Solve. Write answers in standard form. $$ -x(7-2 x)=-6-(3-x) $$

Step-by-Step Solution

Answer

The equation has no real roots.

Step 1: Distribute and simplify both sides

Distribute the \( -x \) across the terms inside the parentheses on the left side of the equation \( -x(7 - 2x) \). This gives \( -7x + 2x^2 \). On the right side, distribute the negative sign across the parentheses \( (3 - x) \) to get \( -6 - 3 + x \). So the equation becomes: \[ 2x^2 - 7x = -9 + x \]

Step 2: Move all terms to one side

Move all terms to one side of the equation to set it to zero. Subtract \( x \) from both sides: \[ 2x^2 - 7x - x = -9 \] which simplifies to: \[ 2x^2 - 8x = -9 \] Now, add 9 to both sides to get: \[ 2x^2 - 8x + 9 = 0 \]

Step 3: Solve the quadratic equation

Now we need to solve the quadratic equation \( 2x^2 - 8x + 9 = 0 \). First, identify \( a = 2 \), \( b = -8 \), and \( c = 9 \). Use the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] to find the roots.

Step 4: Calculate the discriminant

Calculate the discriminant \( b^2 - 4ac \). Substitute the values: \[ (-8)^2 - 4 \cdot 2 \cdot 9 = 64 - 72 = -8 \] Since the discriminant is negative \( (-8) \), this indicates the equation has no real roots.

Key Concepts

Discriminant in Quadratic Equations

When solving quadratic equations like the one in this exercise, the discriminant plays a crucial role in determining the nature of the roots. The discriminant is found in the quadratic formula, specifically under the square root symbol: \( \Delta = b^2 - 4ac\).

Here's why it's important:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is precisely one real root, which means the quadratic has a repeated (or double) root.
• If the discriminant is negative, as in this exercise, the equation has no real roots, indicating that the roots are complex or imaginary.

In our example, calculating the discriminant gave us \((-8)\). The negative value shows that the roots are not real numbers. This outcome helps us understand the possible solutions and directs us on how to interpret or further investigate the given quadratic equation.

Using the Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \[ax^2 + bx + c = 0\].

The formula itself is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

This formula allows you to find the variable \(x\) by simply substituting the coefficients \(a\), \(b\), and \(c\) from the equation. In practice:

• Calculate the discriminant \(b^2 - 4ac\).
• Substitute the discriminant value and coefficients into the formula.
• Use the plus-minus sign to show that there are potentially two solutions.

In the worked example, the solution stops before fully using the quadratic formula because the discriminant was found to be negative. When this happens, it serves as an early signal that the roots are not real and instead are complex. Thus, the formula serves not only as a means to find roots but also as a diagnostic tool to understand the nature of solutions.

Understanding Quadratic Equations in Standard Form

Quadratic equations are often expressed in what is called the "standard form," which is:\[ax^2 + bx + c = 0\].

Here, \(a\), \(b\), and \(c\) are constants, with \(aeq 0\). The standard form is practical because it immediately tells you several things:

• The coefficient \(a\) indicates the parabola's direction: if \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.
• The equation can be easily manipulated for further analysis, such as factoring or applying the quadratic formula.
• Having it in this form helps easily identify the coefficients \(a\), \(b\), and \(c\) needed for computations.

In the exercise, the equation was initially not in standard form. Algebraic manipulation was used to rearrange and simplify terms. This is a typical step in problem-solving to ensure the equation is ready for the application of mathematical formulas or methods, such as using the quadratic formula or graphing to visualize the solutions.