Math  /  Algebra

QuestionCamila Chavez \#5. 1) Solve 2x25x3=02 x^{2}-5 x-3=0 and enter solutions below. 2) Push the "Graph It" Button to see a graph of y=2x25x3y=2 x^{2}-5 x-3
Solution 1: \square
Solution 2: Graph It! \square 10-10

Studdy Solution

STEP 1

1. The equation 2x25x3=0 2x^2 - 5x - 3 = 0 is a quadratic equation.
2. The solutions can be found using the quadratic formula, factoring, or completing the square.
3. The equation can be graphed to visualize the solutions as the x-intercepts.

STEP 2

1. Identify the coefficients for the quadratic formula.
2. Use the quadratic formula to find the solutions.
3. Verify the solutions by substituting back into the original equation.
4. Graph the equation to visualize the solutions.

STEP 3

Identify the coefficients a a , b b , and c c from the quadratic equation 2x25x3=0 2x^2 - 5x - 3 = 0 .
a=2,b=5,c=3 a = 2, \quad b = -5, \quad c = -3

STEP 4

Use the quadratic formula x=b±b24ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} to find the solutions.
Substitute the coefficients into the formula:
x=(5)±(5)242(3)22 x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2}

STEP 5

Calculate the discriminant b24ac b^2 - 4ac :
(5)242(3)=25+24=49 (-5)^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49

STEP 6

Substitute the discriminant back into the quadratic formula:
x=5±494 x = \frac{5 \pm \sqrt{49}}{4}

STEP 7

Calculate the square root and solve for x x :
x=5±74 x = \frac{5 \pm 7}{4}
This gives two solutions:
x=5+74=3 x = \frac{5 + 7}{4} = 3 x=574=12 x = \frac{5 - 7}{4} = -\frac{1}{2}

STEP 8

Verify the solutions by substituting back into the original equation 2x25x3=0 2x^2 - 5x - 3 = 0 .
For x=3 x = 3 :
2(3)25(3)3=18153=0 2(3)^2 - 5(3) - 3 = 18 - 15 - 3 = 0
For x=12 x = -\frac{1}{2} :
2(12)25(12)3=12+523=0 2\left(-\frac{1}{2}\right)^2 - 5\left(-\frac{1}{2}\right) - 3 = \frac{1}{2} + \frac{5}{2} - 3 = 0
Both solutions satisfy the equation.

STEP 9

Graph the equation y=2x25x3 y = 2x^2 - 5x - 3 to visualize the solutions as the x-intercepts at x=3 x = 3 and x=12 x = -\frac{1}{2} .
Solution 1: x=3 x = 3
Solution 2: x=12 x = -\frac{1}{2}

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