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Chapter 6: Problem 33

Solve. See Examples 3 through 6. $$ -3 x^{2}+75=0 $$

Short Answer

Expert verified
The solutions are \(x = 5\) and \(x = -5\).

Step by step solution

01

Isolate the quadratic term

Start by moving the constant term 75 to the other side of the equation by subtracting 75 from both sides. \[-3x^2 + 75 - 75 = 0 - 75\] This simplifies to: \[-3x^2 = -75\]
02

Divide by the coefficient of x虏

Next, divide both sides of the equation by \(-3\) to solve for \(x^2\).\[x^2 = \frac{-75}{-3}\] This simplifies to: \[x^2 = 25\]
03

Solve for x

Take the square root of both sides of the equation to solve for \(x\). Remember, solving \(x^2 = 25\) gives two possible solutions: \[x = \pm \sqrt{25}\]Which simplifies to: \[x = \pm 5\] Thus, the solutions are \(x = 5\) and \(x = -5\).

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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 for solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula provides the solutions to the equation by plugging the coefficients into the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The variables \(a\), \(b\), and \(c\) represent the coefficients of the terms \(x^2\), \(x\), and the constant term, respectively. This formula is very useful when the equation cannot be easily factored.
**Using the Quadratic Formula:**
  • Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
  • Calculate the discriminant \(b^2 - 4ac\).
  • If the discriminant is positive, the equation has two real solutions; if zero, one real solution; and if negative, two complex solutions.
  • Substitute the values into the quadratic formula to find the solutions for \(x\).
By understanding how to use this formula, students can solve any quadratic equation, even if other methods are difficult to apply.
Isolating the Quadratic Term
Isolating the quadratic term is often the first step in solving quadratic equations. The goal is to rewrite the equation so that the \(x^2\) term is by itself on one side of the equation.
In our example, the equation is \(-3x^2 + 75 = 0\). To isolate \(x^2\), you need to:
  • Move other terms to the opposite side of the equation; this usually involves adding or subtracting these terms from both sides.
  • Once other terms are moved, the equation becomes \(-3x^2 = -75\).
By isolating the quadratic term, students set the stage for solving for \(x\) through further operations. This step is crucial as it simplifies the equation and makes the next steps more straightforward.
Taking Square Roots
Taking the square root of both sides of an equation is a common method to solve for \(x\) once the quadratic term is isolated.
In our example, after simplifying to \(x^2 = 25\), you take the square root of both sides: \(x = \pm \sqrt{25}\).
Key points to remember when using this technique:
  • Both positive and negative roots should be considered, meaning \(x = 5\) and \(x = -5\) are both solutions.
  • This method works well when the constant on the other side is a perfect square, which simplifies the calculations.
Taking square roots is a direct method to solve simple quadratics and is extremely efficient when applied correctly. Understanding this concept helps in quickly identifying solutions to quadratic equations.

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Most popular questions from this chapter

Solve each equation. See Examples I and 2. $$ (x+0.2)(x+1.5)=0 $$

Factor out the GCF from each polynomial. See Examples 4 through 10. $$ z(y+4)-3(y+4) $$

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1 ). Don't forget to factor out the GCF first. See Examples I through 10. $$ x^{3} y^{2}+x^{2} y-20 x $$

Multiply the following. See Section 5.4 \(\left(y^{2}+3\right)\left(y^{2}-3\right)\)

Factor each trinomial completely. See Examples I through II and Section 6.2. \(18 x^{2}-9 x-14\)
See all solutions

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