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$6x^2 - 43x - 15 = 0$

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Factor quadratics

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Q.

6x^2 - 43x - 15 = 0

Factor Quadratic Equation: We need to factor the quadratic equation $6x^2 - 43x - 15 = 0$ . To do this, we will look for two numbers that multiply to give the product of the coefficient of $x^2$ (which is $6$ ) and the constant term (which is $-15$ ), and at the same time, these two numbers should add up to give the coefficient of $x$ (which is $-43$ ). $\newline$ Calculation: $6 \times (-15) = -90$ $\newline$ We need to find two numbers that multiply to $-90$ and add up to $-43$ .
Find Suitable Numbers: After trying different combinations, we find that the numbers $-45$ and $2$ satisfy the conditions. They multiply to give $-90$ and add up to give $-43$ . $\newline$ Calculation: $-45 + 2 = -43$ and $-45 \times 2 = -90$
Rewrite Middle Term: Now we rewrite the middle term of the quadratic equation using the two numbers we found. $\newline$ $6x^2 - 45x + 2x - 15 = 0$
Factor by Grouping: Next, we factor by grouping. We group the first two terms and the last two terms. $\newline$ $(6x^2 - 45x) + (2x - 15) = 0$
Factor Out Common Factor: We factor out the greatest common factor from each group. $3x(2x - 15) + 1(2x - 15) = 0$
Set Factors Equal to Zero: We notice that $(2x - 15)$ is a common factor in both groups, so we factor it out. $(2x - 15)(3x + 1) = 0$
Solve for x: Now we have the factored form of the quadratic equation. To find the solutions, we set each factor equal to zero and solve for $x$ . $2x - 15 = 0$ or $3x + 1 = 0$
Solve for x: Now we have the factored form of the quadratic equation. To find the solutions, we set each factor equal to zero and solve for $x$ . $2x - 15 = 0$ or $3x + 1 = 0$ Solving the first equation for $x$ gives us: $2x = 15$ $x = \frac{15}{2}$ $x = 7.5$
Solve for x: Now we have the factored form of the quadratic equation. To find the solutions, we set each factor equal to zero and solve for $x$ . $2x - 15 = 0$ or $3x + 1 = 0$ Solving the first equation for $x$ gives us: $2x = 15$ $x = \frac{15}{2}$ $x = 7.5$ Solving the second equation for $x$ gives us: $3x = -1$ $x = -\frac{1}{3}$

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