$3x^2 +5x + 4 =0$

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Grade 8

Evaluate expressions using properties of exponents

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3x^2 +5x + 4 =0

Identify coefficients: Identify the coefficients of the quadratic equation. $\newline$ The quadratic equation is in the form $ax^2 + bx + c = 0$ . For the equation $3x^2 + 5x + 4 = 0$ , the coefficients are: $a = 3$ , $b = 5$ , and $c = 4$ .
Check factorability: Check if the quadratic equation can be factored easily. $\newline$ In this case, the equation $3x^2 + 5x + 4$ does not factor easily into two binomials. Therefore, we will use the quadratic formula to find the solutions.
Write quadratic formula: Write down the quadratic formula. $\newline$ The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . We will use this formula to find the values of $x$ that satisfy the equation $3x^2 + 5x + 4 = 0$ .
Substitute coefficients: Substitute the coefficients into the quadratic formula. $\newline$ Substitute $a = 3$ , $b = 5$ , and $c = 4$ into the quadratic formula to get $x = \frac{-(5) \pm \sqrt{(5)^2 - 4(3)(4)}}{2(3)}$ .
Simplify and calculate discriminant: Simplify under the square root and calculate the discriminant. $\newline$ Calculate the discriminant $(b^2 - 4ac) = (5)^2 - 4(3)(4) = 25 - 48 = -23$ .
Use complex numbers: Since the discriminant is negative, the solutions will be complex numbers. We can write the solutions as $x = \frac{-5 \pm \sqrt{-23}}{6}$ . The square root of a negative number is an imaginary number, so we can express $\sqrt{-23}$ as $i\sqrt{23}$ , where $i$ is the imaginary unit.
Write final solutions: Write the final solutions. $\newline$ The solutions to the equation $3x^2 + 5x + 4 = 0$ are $x = \frac{-5 + i\sqrt{23}}{6}$ and $x = \frac{-5 - i\sqrt{23}}{6}$ .