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Magazine Chapter 1: Problem 58
Use the quadratic formula and a calculator to find all real solutions, rounded to three decimals. $$ 12.714 x^{2}+7.103 x=0.987 $$
Short Answer
Expert verified
The real solutions are approximately 0.115 and -0.674.
Step by step solution
01
Identify the coefficients
Identify the coefficients from the quadratic equation given: \(ax^2 + bx + c = 0\). In this case, \(a = 12.714\), \(b = 7.103\), and \(c = -0.987\). Note that we bring all terms to one side so the equation becomes: \(12.714x^2 + 7.103x - 0.987 = 0\).
02
Write the quadratic formula
The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula allows us to find the roots of a quadratic equation.
03
Calculate the discriminant
Calculate the discriminant \(\Delta\) using: \(\Delta = b^2 - 4ac\). Substitute the coefficients: \(\Delta = (7.103)^2 - 4(12.714)(-0.987)\).
04
Evaluate the discriminant
Compute \(\Delta = 50.422209 - (-50.123912) = 50.422209 + 50.123912 = 100.546121\). Since the discriminant is positive, there are two real solutions.
05
Apply the quadratic formula
Substitute \(a\), \(b\), and \(\Delta\) into the quadratic formula. Compute both solutions: \[x = \frac{-7.103 \pm \sqrt{100.546121}}{2 \cdot 12.714}\].
06
Calculate the solutions
First compute \( \sqrt{100.546121} \approx 10.025\). Then compute each solution: \(x_1 = \frac{-7.103 + 10.025}{2 \times 12.714} = \frac{2.922}{25.428} \approx 0.115\) and \(x_2 = \frac{-7.103 - 10.025}{2 \times 12.714} = \frac{-17.128}{25.428} \approx -0.674\).
07
Round to three decimals
The final solutions, rounded to three decimal places, are \(x_1 \approx 0.115\) and \(x_2 \approx -0.674\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of any quadratic equation, which is an equation in the form of \( ax^2 + bx + c = 0 \). This formula provides us with a way to find solutions even when factoring is complex or impossible. It looks like this:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's how it works:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here's how it works:
- "\(-b\)" indicates taking the opposite sign of the coefficient in front of \( x \).
- The "\(+\)" and "\(-\)" signs mean you will calculate two different values, leading to two possible solutions for \(x\).
- The term under the square root, "\(b^2 - 4ac\)", is crucial and has a special name: the discriminant.
- Finally, you divide everything by \(2a\) to solve for \(x\).
Discriminant
The discriminant is a key element in understanding the nature of the solutions of a quadratic equation. It is represented by the term \(b^2 - 4ac\) in the quadratic formula. The value of the discriminant tells us how many real solutions the quadratic equation has.
- If the discriminant is positive (\(\Delta > 0\)), there are two distinct real solutions. This is because the square root of a positive number results in two different values when you add and subtract them (hence the \(\pm \) in the formula).
- If the discriminant is zero (\(\Delta = 0\)), there is exactly one real solution because adding and subtracting zero gives the same result.
- If the discriminant is negative (\(\Delta < 0\)), there are no real solutions. Instead, the solutions are complex numbers, which means they include imaginary numbers.
Real Solutions
Real solutions refer to the actual, tangible solutions to a quadratic equation that we can calculate using real numbers. When solving a quadratic equation, the goal is often to find these real solutions.
- We determine real solutions by applying the quadratic formula, only valid when the discriminant is non-negative (zero or positive).
- The plus or minus sign (\(\pm\)) within the formula gives two possibilities, hence potentially two real solutions.
- Each calculation provides a different solution for \(x\), which we found as \(x_1 = 0.115\) and \(x_2 = -0.674\) in the exercise.
- These solutions can be validated by plugging them back into the original equation to see if they satisfy the equation \(12.714x^2 + 7.103x = 0.987\).
