Solve the following quadratic equations.2x2=5x+32x^2=5x+3

x=−12x=-\frac12x=−21 or x=3x=3x=3

Solve the following quadratic equations. 2 x 2 = 5 x + 3 2x^2=5x+3

Here we've been asked to solve this quadratic equation and we're going to do this by factoring. So first, we need to write this equation in standard form and right now it's not. We need to get all the terms on one side. So to do that, we're going to subtract five x and three from both sides so that we have two x squared minus five x minus three equals zero. Now that it's in standard form, we can apply our factoring techniques. Since our coefficient in front of our x squared term is not one, we're going to apply the AC method. So for our AC method, we have a is equal to two and c is equal to negative three, which makes negative six, and our b value is negative five. So we want to find two numbers that multiply to negative six and add to negative five. So let's take a look at one and negative six. These do add to negative five and multiply to negative six, so this is the factor pair we're looking for. Next, we're going to split up our middle term as the sum of these factor pairs. So we can rewrite our equation as two x squared plus x plus negative six x minus three equals zero. And now we can apply factoring by grouping. So we're going to group our first two terms and our last two terms and look for a GCF. For our first two terms, we have a GCF of x and for our last two terms, we have a GCF of negative three. So factoring those out, we have x times two x plus one, plus negative three times two x plus one. And now we have common binomials as well that we can also factor out. Doing that we have two x plus one times x minus three equals zero. Now we're ready to apply our zero product property. So in order to solve this, I want to know when does two x plus one equal zero and when does x minus three equal zero. Well, solving for x in two x plus one, we subtract one from both sides. We have two x equals negative one, then divide by two, and we're left with x equals negative one half. And for x minus three, we add three to both sides and we get x equals three. So these are the two solutions to this equation and we could always check our work by taking these values and plugging them back into our original equation to make sure that it is true.

Solve the following quadratic equations.2x2=$$5x+32x^2=5x+3 $$

x=−12x=-\frac12x=−21 or x=3x=3x=3

x=12x=\frac12x=21 or x=−3x=-3x=−3

x=−12x=-\frac12x=−21 or x=−3x=-3x=−3

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1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
38m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions2h 30m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
54m

12. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

13. Sequences, Series, and the Binomial Theorem1h 50m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

7. Factoring

Quadratic Equations & Applications

Intermediate Algebra

7. Factoring

Quadratic Equations & Applications

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Solve the following quadratic equations.
$2 x^{2} = 5 x + 3$

$x=$\frac$12$ or $x=-3$

$x=-$\frac$12$ or $x=3$

$x=-$\frac$12$ or $x=-3$

$x=$\frac$12$ or $x=3$

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Problema precedenteProssimo problema

Guida verificata passo dopo passo

Rewrite the given equation so that all terms are on one side, setting the equation equal to zero. Starting with $2x^2 = 5x + 3$, subtract \$5x$ and $3$ from both sides to get $2x^2 - 5x - 3 = 0$.

Identify the coefficients in the quadratic equation $ax^2 + bx + c = 0$. Here, $a = 2$, $b = -5$, and $c = -3$.

Use the quadratic formula to solve for $x$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Substitute the values of $a$, $b$, and $c$ into the formula.

Calculate the discriminant $\Delta = b^2 - 4ac$ to determine the nature of the roots. This step helps to know if the solutions are real and distinct, real and equal, or complex.

Simplify the expression under the square root and then simplify the entire formula to find the two possible values for $x$. These will be your solutions to the quadratic equation.