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  1. Hey there, hope I can help!

    [tex]\mathrm{Subtract\:}x^2+5x+39\mathrm{\:from\:both\:sides}[/tex]
    [tex]2x^2+3x-7-\left(x^2+5x+39\right)=x^2+5x+39-\left(x^2+5x+39\right)[/tex]

    Assuming you know how to simplify this, I will not show the steps but can add them later on upon request
    [tex]x^2-2x-46=0[/tex]

    Lets use the quadratic formula now
    [tex]\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}[/tex]
    [tex]x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

    [tex]\mathrm{For\:} a=1,\:b=-2,\:c=-46: x_{1,\:2}=\frac{-\left(-2\right)\pm \sqrt{\left(-2\right)^2-4\cdot \:1\left(-46\right)}}{2\cdot \:1}[/tex]

    [tex]\frac{-\left(-2\right)+\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-46\right)}}{2\cdot \:1} \ \textgreater \ \mathrm{Apply\:rule}\:-\left(-a\right)=a \ \textgreater \ \frac{2+\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-46\right)}}{2\cdot \:1}[/tex]

    Multiply the numbers 2 * 1 = 2
    [tex]\frac{2+\sqrt{\left(-2\right)^2-\left(-46\right)\cdot \:1\cdot \:4}}{2}[/tex]

    [tex]2+\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-46\right)} \ \textgreater \ \sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-46\right)}[/tex]

    [tex]\mathrm{Apply\:rule}\:-\left(-a\right)=a \ \textgreater \ \sqrt{\left(-2\right)^2+1\cdot \:4\cdot \:46} \ \textgreater \ \left(-2\right)^2=2^2, 2^2 = 4[/tex]

    [tex]\mathrm{Multiply\:the\:numbers:}\:4\cdot \:1\cdot \:46=184 \ \textgreater \ \sqrt{4+184} \ \textgreater \ \sqrt{188} \ \textgreater \ 2 + \sqrt{188}[/tex]
    [tex]\frac{2+\sqrt{188}}{2} \ \textgreater \ Prime\;factorize\;188 \ \textgreater \ 2^2\cdot \:47 \ \textgreater \ \sqrt{2^2\cdot \:47}[/tex]

    [tex]\mathrm{Apply\:radical\:rule}: \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b} \ \textgreater \ \sqrt{47}\sqrt{2^2}[/tex]

    [tex]\mathrm{Apply\:radical\:rule}: \sqrt[n]{a^n}=a \ \textgreater \ \sqrt{2^2}=2 \ \textgreater \ 2\sqrt{47} \ \textgreater \ \frac{2+2\sqrt{47}}{2}[/tex]

    [tex]Factor\;2+2\sqrt{47} \ \textgreater \ Rewrite\;as\;1\cdot \:2+2\sqrt{47}[/tex]
    [tex]\mathrm{Factor\:out\:common\:term\:}2 \ \textgreater \ 2\left(1+\sqrt{47}\right) \ \textgreater \ \frac{2\left(1+\sqrt{47}\right)}{2}[/tex]

    [tex]\mathrm{Divide\:the\:numbers:}\:\frac{2}{2}=1 \ \textgreater \ 1+\sqrt{47}[/tex]

    Moving on, I will do the second part excluding the extra details that I had shown previously as from the first portion of the quadratic you can easily see what to do for the second part.

    [tex]\frac{-\left(-2\right)-\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-46\right)}}{2\cdot \:1} \ \textgreater \ \mathrm{Apply\:rule}\:-\left(-a\right)=a \ \textgreater \ \frac{2-\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-46\right)}}{2\cdot \:1}[/tex]

    [tex]\frac{2-\sqrt{\left(-2\right)^2-\left(-46\right)\cdot \:1\cdot \:4}}{2}[/tex]

    [tex]2-\sqrt{\left(-2\right)^2-4\cdot \:1\cdot \left(-46\right)} \ \textgreater \ 2-\sqrt{188} \ \textgreater \ \frac{2-\sqrt{188}}{2}[/tex]

    [tex]\sqrt{188} = 2\sqrt{47} \ \textgreater \ \frac{2-2\sqrt{47}}{2}[/tex]

    [tex]2-2\sqrt{47} \ \textgreater \ 2\left(1-\sqrt{47}\right) \ \textgreater \ \frac{2\left(1-\sqrt{47}\right)}{2} \ \textgreater \ 1-\sqrt{47}[/tex]

    Therefore our final solutions are
    [tex]x=1+\sqrt{47},\:x=1-\sqrt{47}[/tex]

    Hope this helps!

  2. [tex]2x^2 - 3x - 4 = 0 \\ x= \frac{-b \pm \sqrt{ b^{2} -4ac} }{2a} \\ = \frac{-(-3) \pm \sqrt{ (-3)^{2} -4 \times2 \times (-4)} }{2 \times 2} \\ = \frac{3 \pm \sqrt{ 9 +32} }{4} \\ = \frac{3 \pm \sqrt{41} }{4} \\ = \frac{3 + \sqrt{41} }{4} \ or \ \frac{3 - \sqrt{41} }{4} \\ =2.35 \ or \ -0.85[/tex]

  3. Subtract x^2 from both sides
    x^2 + 3x - 7 = 5x + 39
    Subtract 5x from both sides
    x^2 - 2x - 7 = 39
    Add 7 to both sides
    x^2 - 2x = 46
    Complete the square by adding (b/2)^2 to both sides, b = ( -2)
    (-2/2) = -1, then square that (-1)^2 = 1
    x^2 - 2x + 1 = 46 + 1
    Simplify the expression by factoring
    (x - 1)^2 = 47
    Take square root on each side
    x - 1 = (sqrt (47))
    Solve for x
    x = 1 + (sqrt (47))
    Since 47 is prime, 47 cannot be broken down by the square root and this is the answer to your problem.

  4. [tex]x=1\pm\sqrt{47}[/tex]

    Step-by-step explanation:

    We have been given an equation [tex]2x^2+3x-7=x^2+5x+39[/tex]. We are asked to find the solution for our given equation.

    [tex]2x^2+3x-7=x^2+5x+39[/tex]

    [tex]2x^2-x^2+3x-7=x^2-x^2+5x+39[/tex]

    [tex]x^2+3x-7=5x+39[/tex]

    [tex]x^2+3x-5x-7-39=5x-5x+39-39[/tex]

    [tex]x^2-2x-46=0[/tex]

    Using quadratic formula, we will get:

    [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

    [tex]x=\frac{-(-2)\pm\sqrt{(-2)^2-4(1)(-46)}}{2(1)}[/tex]

    [tex]x=\frac{2\pm\sqrt{4+184}}{2}[/tex]

    [tex]x=\frac{2\pm\sqrt{188}}{2}[/tex]

    [tex]x=\frac{2\pm2\sqrt{47}}{2}[/tex]

    [tex]x=1\pm\sqrt{47}[/tex]

    Therefore, the solutions for our given equation are [tex]x=1\pm\sqrt{47}[/tex].

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