Respuesta :
Answer:
[tex]\boxed{4x^2-15x-4=(x-4)(4x+1)}[/tex]
Solution Steps:
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1.) Change the equation using factored transformation:
- [tex]4x^2-15x-4=0[/tex]
- Quadratic polynomial can be factored using the transformation [tex]ax^2+bx+c=a(x-x_{1})(x-x_{2})[/tex], where [tex]x_{1}[/tex] and [tex]x_{2}[/tex] are the solutions of the quadratic equation [tex]ax^2+bx+c=0[/tex].
- This steps basically means change you current equation using the formula [tex]ax^2+bx+c=0[/tex].
2.) Turn the factored form into the quadratic equation form:
- [tex]x=\frac{-(-15)\frac{+}{}\sqrt{(-15)^2-4\bold{x}4(-4)}}{2\bold{x}4}[/tex]
- All equations of the form [tex]ax^2+bx+c=0[/tex] can be solved using the quadratic formula: [tex]\sqrt{\frac{-b\frac{+}{}\sqrt{b^2-4ac}}{2a} }[/tex].
- The quadratic equation formula gives two solutions, one when [tex]\frac{+}{}[/tex] is addition and one when it is subtraction.
3.) Square -15:
- [tex]-15^2=225[/tex]
Equation at the end of Step 3:
- [tex]x=\frac{-(-15)\frac{+}{}\sqrt{225-4\bold{x}4(-4)}}{2\bold{x}4}[/tex]
4.) Multiply −4 times 4:
- [tex]-4[/tex] × [tex]4=-16[/tex]
Equation at the end of Step 4:
- [tex]x=\frac{-(-15)\frac{+}{}\sqrt{225-16(-4)}}{2\bold{x}4}[/tex]
5.) Multiply −16 times −4:
- [tex]-16[/tex] × [tex]-4=64[/tex]
Equation at the end of Step 5:
- [tex]x=\frac{-(-15)\frac{+}{}\sqrt{225+64}}{2\bold{x}4}[/tex]
6.) Add 225 to 64:
- [tex]225+64=289[/tex]
Equation at the end of Step 6:
- [tex]x=\frac{-(-15)\frac{+}{}\sqrt{289}}{2\bold{x}4}[/tex]
7.) Take the square root of 289:
- [tex]\sqrt{289}=17[/tex]
Equation at the end of Step 7:
- [tex]x=\frac{-(-15)\frac{+}{}17}{2\bold{x}4}[/tex]
8.) Change -15 to positive 15:
- [tex]-15=15[/tex]
Equation at the end of Step 8:
- [tex]x=\frac{15\frac{+}{}17}{2\bold{x}4}[/tex]
9.) Multiply 2 by 4:
- [tex]2[/tex] × [tex]4=8[/tex]
Equation at the end of Step 9:
- [tex]x=\frac{15\frac{+}{}17}8}[/tex]
10.) Now Solve:
Now solve the equation [tex]x=\frac{15\frac{+}{}17}8}[/tex] when [tex]\frac{+}{}[/tex] is plus.
Add 15 to 17:
- [tex]15+17=32[/tex]
- [tex]x=\frac{32}{8}[/tex]
Divide 32 by 8:
- [tex]32[/tex] ÷ [tex]8=4[/tex]
- [tex]x=4[/tex]
Now solve the equation [tex]x=\frac{15\frac{+}{}17}8}[/tex] when [tex]\frac{+}{}[/tex] is minus.
Subtract 15 by 17:
- [tex]15-17=-2[/tex]
- [tex]x=\frac{-2}{8}[/tex]
Reduce the fraction to lowest terms by extracting and canceling out 2:
- [tex]-2[/tex] ÷ [tex]-2=-1[/tex]
- [tex]8[/tex] ÷ [tex]-2=-4[/tex]
- [tex]x=-\frac{1}{4}[/tex]
11.) Factor the expression:
Factor the original expression using [tex]ax^2+bx+c=a(x-x_{1})(x-x_{2})[/tex]. Substitute 4 for [tex]x_{1}[/tex] and [tex]-\frac{1}{4}[/tex] for [tex]x_{2}[/tex]:
- [tex]4x^2-15x-4=4(x-4)(x-(-\frac{1}{4}))[/tex]
Simplify all the expressions of the form [tex]p-(-q)[/tex] to [tex]p+q[/tex]:
- [tex]4x^2-15x-4=4(x-4)(x+\frac{1}{4})[/tex]
Add [tex]\frac{1}{4}[/tex] to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible:
- [tex]4x^2-15x-4=4(x-4)\bold{x}(\frac{4x+1}{4})[/tex]
Cancel out 4, the greatest common factor in 4 and 4:
- [tex]4x^2-15x-4=(x-4)(4x+1)[/tex]
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