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Chapter 1: Problem 5

Solve each equation in Exercises \(1-14\) by factoring. $$6 x^{2}+11 x-10=0$$

Short Answer

Expert verified
The solutions to the quadratic equation are \(x = -\frac{5}{2}\) and \(x = \frac{2}{3}\).

Step by step solution

01

Prepare the Equation for Factoring

It is important to ensure that the quadratic equation is in the standard form \( ax^{2} + bx + c = 0 \). The quadratic equation provided is already in standard form, which is \(6x^{2} +11x -10=0\).
02

Factor the Quadratic Equation

The equation is re-written as a product of two binomial expressions. So, it must be arranged like \((px + q)(rx + s) = 0\). Finding the right factors requires that both pairs multiply to \(a \cdot c = -60\) and add up to \(b = 11\). So, the factors will be \(15\) and \(-4\) since they satisfy both conditions. Thus, the factored equation becomes: \(6x^2 -4x +15x -10=0\).
03

Simplify the Factored Equation

The equation can be simplified further by grouping the terms: \(2x(3x -2) +5(3x -2)=0\). Then, you can use the distributive law to condense the equation into its final factored form: \((2x +5)(3x -2)=0\).
04

Solve for \(x\)

By using the Zero-Product Property, which states that if a product of factors equals zero, at least one of the factors must be zero. So, this gives the solutions \(x = -\frac{5}{2}\) or \(x = \frac{2}{3}\).

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Most popular questions from this chapter

In Exercises \(9-20,\) find each product and write the result in standard form. $$-8 i(2 i-7)$$

In Exercises \(9-20,\) find each product and write the result in standard form. $$(2+3 i)^{2}$$

In Exercises \(9-20,\) find each product and write the result in standard form. $$(7-5 i)(-2-3 i)$$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+5}{x-2}>0 $$

Explain how to add complex numbers. Provide an example with your explanation.
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