Problem 46 - Algebra 1

Question

Write the quadratic equation in standard form. Solve using the quadratic formula. $$16=-x^{2}+11 x$$

Step-by-Step Solution

Answer

The standard form of the given equation is \(x^2 - 11x + 16 = 0\). The roots of this equation given by the quadratic formula are \(x = (11 + \sqrt{57}) / 2\) and \(x = (11 - \sqrt{57}) / 2\).

Step 1: Rewrite Equation in Standard Form

Move all terms to one side of the equation to achieve this standard form. Therefore, the equation can be rewritten as \(x^2 - 11x + 16 = 0\).

Step 2: Identify coefficients

The equation \(x^2 - 11x + 16 = 0\) has coefficients a=1, b=-11, and c=16.

Step 3: Apply the Quadratic Formula

Substitute the identified coefficients into the quadratic formula to find the roots of the equation. The quadratic formula is \(x = (-b ± \sqrt{b^2 - 4ac}) / (2a)\). Plugging the coefficients into the formula gives \(x = (11 ± \sqrt{(-11)^2 - 4 * 1 * 16}) / (2 * 1)\). This simplifies to \(x = (11 ± \sqrt{121 - 64}) / 2 = (11 ± \sqrt{57}) / 2\). So the roots of the equation are \(x = (11 + \sqrt{57}) / 2\) and \(x = (11 - \sqrt{57}) / 2\).

Key Concepts

Standard Form

The standard form of a quadratic equation is a way of expressing the equation where all terms are on one side, set equal to zero. For any quadratic equation, standard form is expressed as \( ax^2 + bx + c = 0 \). The coefficients \( a \), \( b \), and \( c \) are crucial as they define the specific nature of the parabola described by the equation.

To convert an equation to standard form, rearrange its terms so that all variable terms and constants are on the same side. Looking at our example, we initially start with the equation \( 16 = -x^{2} + 11x \). For ease of solving, we move all terms to one side:

\( -x^2 + 11x - 16 = 0 \).

Notice, traditionally, we prefer the \( x^2 \) term to be positive in standard form. So, multiplying throughout by \(-1\), we get \( x^2 - 11x + 16 = 0 \). Now, we have successfully written the quadratic equation in its standard form.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation. This formula can solve any quadratic equation, regardless of whether it can be factored easily. The formula is expressed as:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In the formula:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

When you encounter a quadratic equation in standard form, substitute these coefficients into the quadratic formula.

From our example, with the equation \( x^2 - 11x + 16 = 0 \), the coefficients are \( a = 1 \), \( b = -11 \), and \( c = 16 \). Plugging these into our quadratic formula gives us:

\[ x = \frac{11 \pm \sqrt{(-11)^2 - 4 \cdot 1 \cdot 16}}{2 \cdot 1} \]

This calculation involves several steps including squaring the \( b \) coefficient, multiplying \( a \) and \( c \), then subtracting the result from \( b^2 \), which we explore further in the next section.

Roots of the Equation

Finding the roots of a quadratic equation allows us to determine the values of \( x \) where the equation equals zero. These points are where the parabola intersects the x-axis. After substituting our coefficients in the quadratic formula, the next step involves simplifying the expression under the square root, known as the discriminant.

Using our example equation \( x^2 - 11x + 16 = 0 \), the discriminant \( b^2 - 4ac \) is calculated:

\( (-11)^2 - 4 \times 1 \times 16 \), simplifying to \( 121 - 64 \), which then simplifies to \( 57 \).

Since the discriminant is positive, this indicates there are two distinct real roots.

• The calculation becomes \( x = \frac{11 + \sqrt{57}}{2} \)
• And \( x = \frac{11 - \sqrt{57}}{2} \)

This completes the solution. The roots \( \frac{11 + \sqrt{57}}{2} \) and \( \frac{11 - \sqrt{57}}{2} \) represent the x-values where the quadratic equation crosses the x-axis. Understanding these solutions helps in graphing and analyzing the behavior of the quadratic function.