Problem 42 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$-10 a^{2}+3 a+2=0$$

Step-by-Step Solution

Answer

The two possible solutions for \(a\) are \( \frac{3 + \sqrt{89}}{20}\) and \( \frac{3 - \sqrt{89}}{20} \).

Step 1: Identify Your Variables

Let's first identify the coefficients (a, b, c) from the equation. Given the quadratic equation is -10a^2+3a+2=0, then we can identify \(a = -10\), \(b = 3\), and \(c = 2\).

Step 2: Substitute Coefficients into Quadratic Formula

Substitute above values into the quadratic formula: \(-b \pm \sqrt{b^{2} - 4ac} \over 2a\), we get \( -3 \pm \sqrt{(3)^2 - 4*(-10)*2} \over 2*(-10)\).

Step 3: Resolve the Expression

Resolve the expression under the square root first, so it becomes \( -3 \pm \sqrt{9 - (-80)} \over -20 \), after solving it further, it becomes \( -3 \pm \sqrt{89} \over -20 \).

Step 4: Simplifying further

We now have two possible solutions for a: \(a1 = 3 + \sqrt{89} \over 20\), or \(a2 = 3 - \sqrt{89} \over 20\).

Step 5: Evaluating

Evaluating these two results will give the two possible solutions for a.

Key Concepts

Understanding the Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This equation represents a parabola in standard form. The term \(ax^2\) is known as the quadratic term, \(bx\) the linear term, and \(c\) the constant term.
Quadratic equations can appear in various situations, like calculating areas, projectile motion, or optimizing scenarios where the highest or lowest point of a curve is necessary. Solving such equations is fundamental for determining the points at which the parabola intersects the x-axis, known as its roots.

Coefficients Identification in Quadratic Equations

Identifying coefficients is crucial before applying the quadratic formula. In the given equation, \(-10a^2 + 3a + 2 = 0\), you'll identify:

• Coefficient \(a\): the number before \(a^2\), here it is \(-10\).
• Coefficient \(b\): the number before \(a\), here it is \(3\).
• Coefficient \(c\): the constant term, here it is \(2\).

Recognizing these coefficients correctly allows the equation to be solved correctly using the quadratic formula. Misidentification could lead to errors in calculating the roots.

Solving Quadratic Equations Using the Formula

The quadratic formula is a universal method that solves any quadratic equation. It is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula can find the roots or solutions to the quadratic equation by substituting the identified coefficients \(a\), \(b\), and \(c\) into it.
To solve \(-10a^2 + 3a + 2 = 0\), substitute:

• \(a = -10\)
• \(b = 3\)
• \(c = 2\)

We'll substitute these into the formula:
\[a = \frac{-3 \pm \sqrt{3^2 - 4(-10)(2)}}{2(-10)}\]
Carefully conducting the arithmetic operation ensures accurate results when resolving such equations.

Finding Quadratic Roots

Quadratic roots are the solutions for an equation, representing where the parabola crosses the x-axis. Calculating these roots involves taking the expression inside the radical, called the discriminant, \(b^2 - 4ac\). This part determines the number and type of solutions:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If zero, there is exactly one real root (a repeated root).
• If negative, there are two complex roots.

For \(-10a^2 + 3a + 2 = 0\), solving the discriminant gives \(9 - (-80) = 89\), which is positive, meaning there are two real roots. Calculating further gives the results as:
\[a_1 = \frac{3 + \sqrt{89}}{20}\]
\[a_2 = \frac{3 - \sqrt{89}}{20}\]
Thus, understanding the roots is essential in interpreting real-world scenarios modelled by such equations.