Solve the quadratic equation below.
[tex]20 x+14=-3 x^2[/tex]
Give each value of [tex]x[/tex] to 3 significant figures.
Answer (1)
Rewrite the equation in standard form: 3 x 2 + 20 x + 14 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute a = 3 , b = 20 , and c = 14 into the formula and calculate the roots.
Round the solutions to 3 significant figures: x ≈ − 5.87 , x ≈ − 0.795 .
Explanation
Understanding the Problem We are given the quadratic equation 20 x + 14 = − 3 x 2 . Our goal is to solve for x and express the solutions to 3 significant figures. To do this, we will first rewrite the equation in the standard quadratic form, identify the coefficients, and then apply the quadratic formula.
Rewriting the Equation First, we rewrite the given equation in the standard quadratic form a x 2 + b x + c = 0 . Adding 3 x 2 to both sides of the equation 20 x + 14 = − 3 x 2 , we get 3 x 2 + 20 x + 14 = 0 .
Identifying Coefficients Now, we identify the coefficients a , b , and c in the quadratic equation 3 x 2 + 20 x + 14 = 0 . We have a = 3 , b = 20 , and c = 14 .
Applying the Quadratic Formula Next, we use the quadratic formula to find the solutions for x . The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c .
Substituting Values We substitute the values of a , b , and c into the quadratic formula: x = 2 ( 3 ) − 20 ± 2 0 2 − 4 ( 3 ) ( 14 ) x = 6 − 20 ± 400 − 168 x = 6 − 20 ± 232
Calculating the Roots Now, we calculate the two possible values of x :
x 1 = 6 − 20 − 232 ≈ 6 − 20 − 15.23 ≈ 6 − 35.23 ≈ − 5.8719 x 2 = 6 − 20 + 232 ≈ 6 − 20 + 15.23 ≈ 6 − 4.77 ≈ − 0.795
Rounding to 3 Significant Figures Finally, we round each solution to 3 significant figures: x 1 ≈ − 5.87 x 2 ≈ − 0.795
Final Answer Therefore, the solutions to the quadratic equation 20 x + 14 = − 3 x 2 , rounded to 3 significant figures, are x ≈ − 5.87 and x ≈ − 0.795 .
Examples
Quadratic equations are incredibly useful in physics, engineering, and economics. For example, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch, ensuring it can withstand specific loads and stresses. Similarly, in economics, quadratic functions can model cost and revenue curves, helping businesses determine the optimal production level to maximize profit. Understanding how to solve quadratic equations allows professionals to make informed decisions and create efficient designs.
