Problem 51 - Algebra 1

Question

Write the quadratic equation in standard form. Solve using the quadratic formula. $$-5 c^{2}+9 c=4$$

Step-by-Step Solution

Answer

The roots for the quadratic equation \(-5c^{2} + 9c - 4 = 0\) are \(c1 = 0.8\) and \(c2 = -1\).

Step 1: Write in Standard Form

Rearrange the given equation to follow the standard form. So, the equation becomes \(-5c^{2} + 9c - 4 = 0\).

Step 2: Apply the Quadratic Formula

Now, use the quadratic formula to determine the solutions. Substitute \(a = -5\), \(b = 9\), and \(c = -4\) into \(-\frac{b \pm \sqrt{b^{2} - 4ac}}{2a}\). This gives \(-\frac{9 \pm \sqrt{(9)^{2} - 4(-5)(-4)}}{2(-5)} = -\frac{9 \pm \sqrt{81 - 80}}{-10}\).

Step 3: Simplify the equation

Simplify the equation to find the solutions. The simplified equation becomes \(-\frac{9 \pm \sqrt{1}}{-10} = -\frac{9 \pm 1}{-10}\). Therefore, the roots of the equation are \(c1 = \frac{9 - 1}{10} = 0.8\) and \(c2 = \frac{9 + 1}{10} = -1\).

Key Concepts

Standard Form of Quadratic Equation

Quadratic equations are fundamental to algebra, and their standard form is pivotal for solving them. The standard form of a quadratic equation is written as:
\[ ax^2 + bx + c = 0 \]
The letters \( a \), \( b \), and \( c \) represent coefficients, with \( a \) being the coefficient of the squared term and it cannot be zero. When you're given a quadratic expression such as \( -5c^2 + 9c = 4 \), the first step is to rearrange it to this standard form. By doing so, you set the stage for finding the equation's roots using various methods, including the renowned quadratic formula.
Moving the constant term to the left side by subtracting 4 from both sides, \( -5c^2 + 9c - 4 = 0 \), achieves the standard form, laying the groundwork for the next step: solving the equation.

Solving Quadratic Equations

Solving quadratic equations is a skill that unlocks the potential to tackle a myriad of problems in mathematics. Once in standard form, a range of techniques can be employed to find the roots of the quadratic equation, but one of the most efficient and universally applicable methods is the quadratic formula.

Quadratic Formula

The quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
provides an axiom to calculate the roots of any quadratic equation. To use this formula, simply identify the coefficients \( a \), \( b \), and \( c \) from the standard form of the equation. In the exercise, we have \( a = -5 \), \( b = 9 \), and \( c = -4 \). Plugging these values into the formula gives two possible values for \( c \), known as the roots. The process of substitution and simplifying leads to the solution of the quadratic equation.

Quadratic Roots

Quadratic roots, also known as the solutions or zeros of the equation, are the values of \( x \) (or in our exercise, \( c \)) that satisfy the equation \( ax^2 + bx + c = 0 \). There can be two roots, one root, or no real roots, depending on the discriminant, \( b^2 - 4ac \).

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is one real root.
• If the discriminant is negative, there are no real roots, but two complex roots.

In our given problem, after using the quadratic formula and simplifying, we find two real roots: \( c1 = 0.8 \) and \( c2 = -1\). These roots are where the parabola, the graph of the quadratic equation, intersects the horizontal axis, providing critical points for graphing and understanding the behavior of quadratic functions.