# Recent questions tagged roots

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Given: $$\quad X=\sqrt{e-4}$$Write down the value of $$e$$, for which $$X$$ is:1. Zero2. Non-realDetermine the value(s) of $$m$$ for which the equation $$m x^2-12 x+9=0$$...
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Find the point(s) on the parabola $$2 x=y^{2}$$ closest to the point $$(1,0)$$.
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Given the equation $$4(x+1)=3x+10$$. For which values of $$x$$ is the equation true?
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For which values of $$k$$ will the equation $$x^2+6 x-2 k=0$$ have non-real roots?
666
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Calculate the roots of the cubic equation $$x^3-8=0$$, show all working. Use latex.
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Give an example of how to find the roots of a complex number. use latex
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Without solving for $$x$$, determine the nature of the roots of the following equation:$x^2-4 x=-3$
508
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Given that $$2 x^2+(p-3) x+8=0$$, determine the value(s) of $$p$$ for which the roots will be non-real.
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The roots of a quadratic equation are given as $$x=\dfrac{-2 \pm \sqrt{1-3 k}}{4}$$.a) For which value(s) of $$k$$ will the roots be real?b) For which value(s) of $$k$$ w...
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If $$a^2(c-b) ; a(c-b) ; b-a$$ forms a geometric sequence, prove that $$a ; b ; c$$ forms an arithmetic sequence.
411
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If $$a x^2+2 b x+c=0$$ has equal roots, prove that $$a ; b ; c$$ forms a geometric sequence.
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The roots of the polynomial $$x^4-4 i x^3+3 x^2-14 i x-44$$ form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram?
359
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Let $$r, s, t$$ be the roots of the equation $$x(x-2)(3 x-7)=2$$. Show that $$r, s, t$$ are real and positive and determine $$\arctan r+\arctan s+\arctan t$$.
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Find the roots $$r_1, r_2, r_3, r_4$$ of the equation $$4 x^4-a x^3+b x^2-c x+5=0$$, assuming that they are positive real numbers and that$\frac{r_1}{2}+\frac{r_2}{4}+\f... 274 views 0 answers Find the number of real roots of the equation\[x^8-x^7+2 x^6-2 x^5+3 x^4-3 x^3+4 x^2-4 x+\frac{5}{2}=0 .$
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Find the number of real roots of the equation$x^8-x^7+2 x^6-2 x^5+3 x^4-3 x^3+4 x^2-4 x+\frac{5}{2}=0 .$
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(a) Find the roots of the equation$z^2+2 z+17=0,$giving your answers in the form $$a+\mathrm{i} b$$, where $$a$$ and $$b$$ are integers.(b) Show these roots on an Argan...
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(a) Write down the value of the real root of the equation$x^3-64=0$(b) Find the complex roots of $$x^3-64=0$$, giving your answers in the form $$a+\mathrm{i} b$$, where...
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$f(x)=x^3+x^2+44 x+150$Given that $$\mathrm{f}(x)=(x+3)\left(x^2+a x+b\right)$$, where $$a$$ and $$b$$ are real constants,(a) find the value of $$a$$ and the value of \...
267
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How do I determine squares, square roots, cubes and cube roots of algebraic terms?
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Can i determine the nature of roots of an equation or function without doing the calculation?
362
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The number of distinct terms in $$(a+b+c+d+e)^{4}$$ is
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$x^{2}+2 x+3=0$if a is one of its roots, then$\frac{a^{5}+3 a^{4}+3 a^{3}-a^{2}}{a^{2}+3}=?$
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Find the unique value of $$a$$ for which the polynomial $$x^{3}-12 x^{2}+a x-64$$ has all real, nonnegative roots.
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Find all the integer roots of $$2 x^{4}+4 x^{3}-5 x^{2}+2 x-3=0$$. Enter all the integer roots, separated by commas.
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Find all the integer roots of$x^{4}+5 x^{3}+9 x^{2}-x-14=0 .$Enter all the integer roots, separated by commas.
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Let $$a, b, c, d$$, and $$e$$ be the distinct roots of the equation $$x^{5}+7 x^{4}-2=0$$. Find\begin{aligned}&\frac{a^{3}}{(a-b)(a-c)(a-d)(a-e)}+\frac{b^{3}}{(b-a)(b-c... 317 views 1 answers For how many different values of $$x$$ does $$x^{5}=x^{4}+72 x^{3}$$ ? 355 views 1 answers If\[C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0where $$C_{0}, \ldots, C_{n}$$ are real constants, prove that the equation$C_{0}+C_{1} x+\cdots+C... 262 views 1 answers Prove that $$\sum 1 / p$$ diverges; the sum extends over all primes.(This shows that the primes form a fairly substantial subset of the positive integers.)Hint: Given $$N... 178 views 1 answers Solve \(5x^2-49=0$$ 428 views 1 answers If $${ }^{n} P_{r}:^{n} C_{r}=120$$ then find $$n$$ and $$r$$ 427 views 1 answers Show that $$x^{2}+2 x^{2}+5=0$$ has non-real roots 288 views 0 answers $$\mathrm{a}, \mathrm{b}, \ldots, \mathrm{e}$$ are positive integersabcde $$=a+b+c+d+e$$$$\max (\max (a, b, c, d, e))=?$$ 624 views 0 answers For what values of the constant $$c$$ does $$\ln x=c x^{2}$$ have solutions. Assume that $$c>0$$. 564 views 0 answers Find a curve $$y=f(x)$$ with the following properties:- $$\frac{d^{2} y}{d x^{2}}=6 x$$- Its graph passes through the point $$(0,1)$$ and has a horizontal tangent there. 940 views 1 answers Find, correct to 5 decimal places, the $$x$$-coordinate of the point on the curve $$y=\ln x$$ which is closest to the origin. Use the Newton Method. 1.6k views 1 answers Use the Newton Method to find the smallest and the second smallest positive roots of the equation $$\tan x=4 x$$, correct to 4 decimal places. 1.3k views 1 answers A calculator is defective: it can only add, subtract, and multiply. Use the equation $$1 / x=1.37$$, the Newton Method, and the defective calculator to find $$1 / 1.37$$ ... 1.3k views 1 answers Find all solutions of $$5 x+\ln x=10000$$, correct to 4 decimal places; use the Newton Method. 1.5k views 1 answers Find all solutions of $$e^{2 x}=x+6$$, correct to 4 decimal places; use the Newton Method. 259 views 1 answers Solve the equation:$$5(2-3x)+3(5x-2)-7(4-x)=0$$ 517 views 1 answers Solve the proportion $$\dfrac{5}{3}=\dfrac{x}{6}$$ for the unknown value $$x$$. 353 views 1 answers Prove that, for all values of $$x$$,\[x^{2}+4 x+12>x+3$
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Show that $$x^{2}-6 x+11>0$$ for all real values of $$x$$.
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If $$x^{2}-k x+9=0$$ has equal roots, find the values of $$k$$.A. 3,4B. $$\pm 3$$C. $$\pm 5$$D. $$\pm 6$$
How would you solve the equation $$\dfrac{(v+2)^{2}-6}{3}=1$$ ?