Example:
[tex]\begin{array}{l}
-2 x^2+4 x+70=0 \\
-2\left(x^2-\ldots x-\ldots\right)=0 \\
-2(x-\ldots)(x+\ldots)=0 \\
x-\ldots=0 \text { and } x+\ldots= \\
x=\ldots \text { and } x=
\end{array}[/tex]
Answer (2)
To solve the quadratic equation − 2 x 2 + 4 x + 70 = 0 , we factor out − 2 to get − 2 ( x 2 − 2 x − 35 ) = 0 . Factoring the quadratic gives us − 2 ( x − 7 ) ( x + 5 ) = 0 , leading to solutions of x = 7 and x = − 5 .
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Factor out the common factor: − 2 ( x 2 − 2 x − 35 ) = 0 .
Factor the quadratic expression: − 2 ( x − 7 ) ( x + 5 ) = 0 .
Set each factor to zero: x − 7 = 0 and x + 5 = 0 .
Solve for x : x = 7 and x = − 5 . The solutions are x = 7 and x = − 5 .
Explanation
Problem Analysis We are given the quadratic equation − 2 x 2 + 4 x + 70 = 0 and we need to solve for x by factoring. Let's walk through the process step by step.
Factoring out -2 First, we factor out the common factor of -2 from the entire equation: − 2 ( x 2 − 2 x − 35 ) = 0
Factoring the Quadratic Now, we need to factor the quadratic expression x 2 − 2 x − 35 . We are looking for two numbers that multiply to -35 and add to -2. These numbers are -7 and +5. So, we can write the quadratic expression as: ( x − 7 ) ( x + 5 )
Complete Factored Form Putting it all together, the factored form of the original equation is: − 2 ( x − 7 ) ( x + 5 ) = 0
Setting Factors to Zero To find the solutions for x , we set each factor containing x equal to zero: x − 7 = 0 and x + 5 = 0
Solving for x Solving these equations for x , we get: x = 7 and x = − 5 So, the solutions to the quadratic equation are x = 7 and x = − 5 .
Final Answer Therefore, the final answer is x = 7 and x = − 5 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth or decay of populations. For instance, if you're designing a bridge, you might use a quadratic equation to model the curve of an arch, ensuring it can withstand specific loads and stresses. Understanding how to solve quadratic equations helps in making informed decisions in engineering, physics, economics, and many other fields.
