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 \(m x^2-12 x+9=0\)...
130
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Find the point(s) on the parabola \( 2 x=y^{2} \) closest to the point \( (1,0) \).
170
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Given the equation \(4(x+1)=3x+10\). For which values of \(x\) is the equation true?
224
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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.
318
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Give an example of how to find the roots of a complex number. use latex
206
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467
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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.
381
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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...
529
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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.
202
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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 \(\arctan r+\arctan s+\arctan t\).
240
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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
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Find the number of real roots of the equation\[x^8-x^7+2 x^6-2 x^5+3 x^4-3 x^3+4 x^2-4 x+\frac{5}{2}=0 .\]
286
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Find the number of real roots of the equation\[x^8-x^7+2 x^6-2 x^5+3 x^4-3 x^3+4 x^2-4 x+\frac{5}{2}=0 .\]
513
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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...
211
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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...
277
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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?
346
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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
621
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\[x^{2}+2 x+3=0\]if a is one of its roots, then\[\frac{a^{5}+3 a^{4}+3 a^{3}-a^{2}}{a^{2}+3}=?\]
229
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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.
253
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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.
271
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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.
502
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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
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1 answers
For how many different values of \(x\) does \(x^{5}=x^{4}+72 x^{3}\) ?
355
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If\[C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0\]where \(C_{0}, \ldots, C_{n}\) are real constants, prove that the equation\[C_{0}+C_{1} x+\cdots+C...
262
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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
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Solve \(5x^2-49=0 \)
428
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If \({ }^{n} P_{r}:^{n} C_{r}=120\) then find \(n\) and \(r\)
427
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Show that \(x^{2}+2 x^{2}+5=0\) has non-real roots
288
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\(\mathrm{a}, \mathrm{b}, \ldots, \mathrm{e}\) are positive integersabcde \(=a+b+c+d+e\)\(\max (\max (a, b, c, d, e))=?\)
624
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For what values of the constant \(c\) does \(\ln x=c x^{2}\) have solutions. Assume that \(c>0\).
564
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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
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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
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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
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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
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Find all solutions of \(5 x+\ln x=10000\), correct to 4 decimal places; use the Newton Method.
1.5k
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Find all solutions of \(e^{2 x}=x+6\), correct to 4 decimal places; use the Newton Method.
259
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Solve the equation:\(5(2-3x)+3(5x-2)-7(4-x)=0\)
517
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Solve the proportion \(\dfrac{5}{3}=\dfrac{x}{6}\) for the unknown value \(x\).
353
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Prove that, for all values of \(x\),\[x^{2}+4 x+12>x+3\]
662
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Show that \(x^{2}-6 x+11>0\) for all real values of \(x\).
437
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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. \(\pm 6\)
410
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What is the prime factorization of 136?
320
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How would you solve the equation \(\dfrac{(v+2)^{2}-6}{3}=1\) ?
499
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How do you multiply two matrices?
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