Problem 25 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ -2 t(t+2)=-3 $$

Step-by-Step Solution

Answer

The solutions are \( t = -1 + \frac{\sqrt{10}}{2} \) and \( t = -1 - \frac{\sqrt{10}}{2} \)

Step 1: Expand the left side

Distribute \(-2 t(t + 2)\). This gives: \(-2 t^2 - 4 t\)

Step 2: Move all terms to one side

Add \(-3\) to both sides to form a standard quadratic equation: \(-2 t^2 - 4 t + 3 = 0\)

Step 3: Identify coefficients

Identify the coefficients \a=-2\, \b=-4\, and \c=3\ from the equation \(-2 t^2 - 4 t + 3 = 0\).

Step 4: Write the quadratic formula

The quadratic formula is given by: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 5: Substitute the coefficients into the quadratic formula

Substitute \a=-2\, \b=-4\, and \c=3\ into the quadratic formula: \[ t = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-2)(3)}}{2(-2)} \]

Step 6: Simplify inside the square root

Calculate the discriminant: \[ (-4)^2 - 4(-2)(3) = 16 + 24 = 40 \]

Step 7: Solve for t

Replace the discriminant and simplify further: \[ t = \frac{4 \pm \sqrt{40}}{-4} \] Simplify \sqrt{40} = 2\sqrt{10}\ and get \[ t = \frac{4 \pm 2 \sqrt{10}}{-4} \] Then split into two solutions: \[ t = \frac{2 \pm \sqrt{10}}{-2} = -1 \mp \frac{\sqrt{10}}{2} \]

Step 8: Write the final solutions

The solutions are: \[ t = -1 + \frac{\sqrt{10}}{2} \] and \[ t = -1 - \frac{\sqrt{10}}{2} \]

Key Concepts

quadratic formula

The quadratic formula provides a foolproof method for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). Instead of factoring or graphing, it uses the coefficients directly to find the solutions. The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

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

When you apply these values, the formula will give you two solutions (roots) for the quadratic equation, thanks to the \(\pm\) symbol that represents addition and subtraction.
Steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from your equation.
• Plug these values into the formula.
• Simplify the expression to find the roots.

discriminant calculation

One of the critical parts of the quadratic formula is the discriminant, which is the expression under the square root: \(b^2 - 4ac\). The discriminant tells you about the nature of the roots of the quadratic equation. Here's what it reveals:

• If \(b^2 - 4ac > 0\), there are two real and distinct solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution (a repeated root).
• If \(b^2 - 4ac < 0\), there are no real solutions (the solutions are complex or imaginary).

To perform the discriminant calculation, simply:

• Square the coefficient \(b\).
• Multiply \(4\) by \(a\) and \(c\).
• Subtract the product from \(b^2\).

For our equation:\[(-4)^2 - 4(-2)(3) = 16 + 24 = 40\]We find that the discriminant is \(40\), which is positive and indicates two distinct real solutions.

simplifying radicals

When using the quadratic formula, you often end up with a square root that needs simplifying. Simplifying radicals makes your final answer cleaner and easier to interpret.

First, find the prime factors of the number under the square root (called the radicand). For example, if you need to simplify \(\sqrt{40}\):

• Break \(40\) into its prime factors: \(40 = 2^3 \times 5\).
• Rewrite the square root using these factors: \(\sqrt{40} = \sqrt{2^3 \times 5}\).
• Simplify: \(\sqrt{2^3} = 2\sqrt{10} \Rightarrow \sqrt{40} = 2\sqrt{10}\).

Putting this into our quadratic formula solution:\[t = \frac{4 \pm 2\sqrt{10}}{-4}\]Finally, simplify the fraction:

• Divide both the numerator and the denominator by \(2\): \[t = \frac{2 \pm \sqrt{10}}{-2}\]
• Split the equation into two parts: \[t = -1 \mp \frac{\sqrt{10}}{2}\]

So, the solutions to the quadratic equation are:

• \(t = -1 + \frac{\sqrt{10}}{2}\)
• \(t = -1 - \frac{\sqrt{10}}{2}\)