Problem 62 - Algebra 2

Question

Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. $$ 4 x^{2}+7 x=11 $$

Step-by-Step Solution

Answer

The discriminant is 225; the equation has two distinct real roots.

Step 1: Rewrite the Equation in Standard Form

First, we need to rewrite the given equation in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). The given equation is \(4x^2 + 7x = 11\), so we subtract 11 from both sides of the equation to get \(4x^2 + 7x - 11 = 0\).

Step 2: Identify Coefficients a, b, and c

Now that the equation is in standard form, identify the coefficients: \(a = 4\), \(b = 7\), \(c = -11\). These values will be used to calculate the discriminant.

Step 3: Calculate the Discriminant

The formula for the discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is \(b^2 - 4ac\). Substitute the identified values into the formula: \(\Delta = 7^2 - 4 \times 4 \times (-11)\). Calculate \(\Delta = 49 + 176\). Thus, \(\Delta = 225\).

Step 4: Interpret the Discriminant

The value of the discriminant is \(\Delta = 225\). Since \(\Delta > 0\), the quadratic equation has two distinct real roots.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. It has the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known as the coefficients. The variable \(x\) is what we're solving for.
The key characteristic of a quadratic equation is the "square" term (\(x^2\)). This term is what makes the equation quadratic. Without it, the equation would only be linear. Since it's of degree two, a quadratic equation can have up to two solutions, or roots. These roots are the values of \(x\) that satisfy the equation.
Solving a quadratic equation typically involves finding these roots. Methods like factoring, completing the square, or utilizing the quadratic formula are common approaches. Each method aims to isolate \(x\) and solve for its value by making the equation equal to zero.

Real Roots

Real roots of a quadratic equation are the solutions that are real numbers. This means they are not imaginary or complex, and can be plotted on the real number line. In context of the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), the expression \(b^2 - 4ac\) is crucial since it determines the nature of the roots.
This expression is known as the "discriminant". A positive discriminant indicates two distinct real roots. A zero discriminant results in one real root, also known as a repeated or double root. Lastly, a negative discriminant signifies no real roots but two complex conjugates.
In the exercise, after calculating the discriminant, it was found to be 225, which is positive. This confirms that the quadratic equation indeed has two distinct real roots.

Coefficients

Coefficients are the numerical or constant multipliers of the variables in an equation. In the quadratic equation \(ax^2 + bx + c = 0\), the coefficients are \(a\), \(b\), and \(c\). They help determine the shape and position of the parabola when the equation is graphed.
Each coefficient plays a distinct role:

• \(a\): The leading coefficient, it affects the direction (upward or downward) and the steepness of the parabola. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
• \(b\): This coefficient influences the position of the vertex along the x-axis, impacting symmetry.
• \(c\): This constant term determines the point where the parabola intersects the y-axis.

In the given exercise, the coefficients were \(a = 4\), \(b = 7\), and \(c = -11\). Using these, the discriminant was calculated to determine the number and type of roots.