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Magazine Problem 62
Determine all of the real-number solutions for each equation. (Remember to check for extraneous solutions.) $$\sqrt{x^{2}+5 x-2}=2$$
Problem 63
Show that the quadratic equation $$a x^{2}+b x-a=0 \quad(a \neq 0)$$ has two distinct real roots.
Problem 63
Determine all of the real-number solutions for each equation. (Remember to check for extraneous solutions.) $$\sqrt{x}+6=x$$
Problem 64
Determine all of the real-number solutions for each equation. (Remember to check for extraneous solutions.) $$x-\sqrt{x}=20$$
Problem 64
Show that the quadratic equation $$(x-p)(x-q)=r^{2} \quad(p \neq q)$$ has two distinct real roots.
Problem 65
Determine the value(s) of the constant \(k\) for which the equation has equal roots (that is, only one distinct root). $$x^{2}=2 x(3 k+1)-7(2 k+3)$$
Problem 65
Determine all of the real-number solutions for each equation. (Remember to check for extraneous solutions.) $$x-\sqrt{3-x}=-3$$
Problem 65
(a) Use a graph to estimate the solution set for each inequality. Zoom in far enough so that you can estimate the relevant endpoints to the nearest thousandth. (b) Exercises \(61-70\) can be solved algebraically using the techniques presented in this section. Carry out the algebra to obtain exact expressions for the endpoints that you estimated in part (a). Then use a calculator to check that your results are consistent with the previous estimates. $$x^{4}-2 x^{2}-1>0$$
Problem 66
Determine all of the real-number solutions for each equation. (Remember to check for extraneous solutions.) $$\sqrt{2-x}-10=x$$
Problem 66
Determine the value(s) of the constant \(k\) for which the equation has equal roots (that is, only one distinct root). $$x^{2}+2(k+1) x+k^{2}=0$$
